{"id":110,"date":"2025-04-09T17:10:03","date_gmt":"2025-04-09T17:10:03","guid":{"rendered":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/chapter\/2-4-complex-numbers-college-algebra-2e-openstax\/"},"modified":"2026-03-19T15:56:31","modified_gmt":"2026-03-19T15:56:31","slug":"2-4-complex-numbers","status":"publish","type":"chapter","link":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/chapter\/2-4-complex-numbers\/","title":{"raw":"2.4 Complex Numbers","rendered":"2.4 Complex Numbers"},"content":{"raw":"<div id=\"main-content\" class=\"MainContent__ContentStyles-sc-6yy1if-0 NnXKu\" tabindex=\"-1\" data-dynamic-style=\"true\">\r\n<div id=\"page_4aa9351c-019f-4c06-bb40-d58262ea7ec7\" class=\"chapter-content-module\" data-type=\"page\" data-book-content=\"true\">\r\n<div class=\"ui-has-child-title\" data-type=\"abstract\"><header>\r\n<h2 data-type=\"title\">Learning Objectives<\/h2>\r\n<\/header><section>\r\n<p id=\"para-00001\">In this section, you will:<\/p>\r\n\r\n<ul id=\"list-00001\">\r\n \t<li>Add and subtract complex numbers.<\/li>\r\n \t<li>Multiply and divide complex numbers.<\/li>\r\n \t<li>Simplify powers of\r\n<math display=\"inline\"><semantics><mrow><mi>i<\/mi><\/mrow><annotation-xml encoding=\"MathML-Content\"><mi>i<\/mi><\/annotation-xml><\/semantics><\/math><\/li>\r\n<\/ul>\r\n<\/section><\/div>\r\n<div id=\"Figure_02_04_001\" class=\"os-figure\">\r\n<figure class=\"medium\" data-id=\"Figure_02_04_001\"><span id=\"fs-id1919899\" data-type=\"media\" data-alt=\"A visual representation of the Mandelbrot set\">\r\n<img src=\"https:\/\/openstax.org\/books\/college-algebra-2e\/pages\/src#fixme\" alt=\"A visual representation of the Mandelbrot set\" width=\"437\" height=\"325\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/250825d36367ccdee3e00cbf20958515aa106674\" \/>\r\n<\/span><\/figure>\r\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Figure <\/span>\r\n<span class=\"os-number\">1<\/span><\/div>\r\n<\/div>\r\n<p id=\"fs-id2523388\">Discovered by Benoit Mandelbrot around 1980, the Mandelbrot Set is one of the most recognizable fractal images. The image is built on the theory of self-similarity and the operation of iteration. Zooming in on a fractal image brings many surprises, particularly in the high level of repetition of detail that appears as magnification increases. The equation that generates this image turns out to be rather simple.<\/p>\r\n<p id=\"fs-id2510203\">In order to better understand it, we need to become familiar with a new set of numbers. Keep in mind that the study of mathematics continuously builds upon itself. Negative integers, for example, fill a void left by the set of positive integers. The set of rational numbers, in turn, fills a void left by the set of integers. The set of real numbers fills a void left by the set of rational numbers. Not surprisingly, the set of real numbers has voids as well. In this section, we will explore a set of numbers that fills voids in the set of real numbers and find out how to work within it.<\/p>\r\n\r\n<section id=\"fs-id2425607\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Expressing Square Roots of Negative Numbers as Multiples of <math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>i<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>i<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/h2>\r\n<p id=\"fs-id1385456\">We know how to find the square root of any positive real number. In a similar way, we can find the square root of any negative number. The difference is that the root is not real. If the value in the radicand is negative, the root is said to be an imaginary number<strong>. <\/strong>The imaginary number<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>i<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>i<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>is defined as the square root of\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>\u22121.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>\u22121.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n<div id=\"fs-id1276418\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>\u22121<\/mn>\r\n<\/mrow>\r\n<\/msqrt>\r\n<mo>=<\/mo><mi>i<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>\u22121<\/mn>\r\n<\/mrow>\r\n<\/msqrt>\r\n<mo>=<\/mo><mi>i<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id3155144\">So, using properties of radicals,<\/p>\r\n\r\n<div id=\"fs-id2005251\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mo>\u2212<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<\/msqrt>&nbsp;\r\n\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><mn>\u22121<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mo>\u2212<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<\/msqrt>&nbsp;\r\n\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><mn>\u22121<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1523248\">We can write the square root of any negative number as a multiple of<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>i<\/mi><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>i<\/mi><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>Consider the square root of\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>\u221249.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>\u221249.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n<div id=\"fs-id1296283\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><msqrt><mrow><mn>\u221249<\/mn><\/mrow><\/msqrt><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>49<\/mn><mo>\u22c5<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u22121<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>49<\/mn><\/mrow>\r\n<\/msqrt>\r\n<msqrt>\r\n<mrow>\r\n<mn>\u22121<\/mn><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>7<\/mn><mi>i<\/mi><\/mrow><\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><msqrt><mrow><mn>\u221249<\/mn><\/mrow><\/msqrt><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>49<\/mn><mo>\u22c5<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u22121<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>49<\/mn><\/mrow>\r\n<\/msqrt>\r\n<msqrt>\r\n<mrow>\r\n<mn>\u22121<\/mn><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>7<\/mn><mi>i<\/mi><\/mrow><\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1149119\">We use<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>7<\/mn><mi>i<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>7<\/mn><mi>i<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>and not\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>\u22127<\/mn><mi>i<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>\u22127<\/mn><mi>i<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>because the principal root of\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>49<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>49<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>is the positive root.\r\n<p id=\"fs-id2708516\">A complex number is the sum of a real number and an imaginary number. A complex number is expressed in standard form when written<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>where\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>a<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>a<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>is the real part and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>b<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>b<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>is the imaginary part. For example,\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>5<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>5<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>is a complex number. So, too, is\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>3<\/mn><mo>+<\/mo><mn>4<\/mn><mi>i<\/mi><msqrt>\r\n<mn>3<\/mn>\r\n<\/msqrt>\r\n<mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>3<\/mn><mo>+<\/mo><mn>4<\/mn><mi>i<\/mi><msqrt>\r\n<mn>3<\/mn>\r\n<\/msqrt>\r\n<mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<span id=\"fs-id1541885\" data-type=\"media\" data-alt=\"The complex number 5 + 2i is displayed. The 5 is labeled as: Real part and the 2i is labeled as: Imaginary part\">\r\n<img src=\"https:\/\/openstax.org\/books\/college-algebra-2e\/pages\/src#fixme\" alt=\"The complex number 5 + 2i is displayed. The 5 is labeled as: Real part and the 2i is labeled as: Imaginary part\" width=\"487\" height=\"76\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/b6771cfacdac806d7546b9b1d376853f0f822803\" \/>\r\n<\/span>\r\n<p id=\"fs-id1174676\">Imaginary numbers differ from real numbers in that a squared imaginary number produces a negative real number. Recall that when a positive real number is squared, the result is a positive real number and when a negative real number is squared, the result is also a positive real number. Complex numbers consist of real and imaginary numbers.<\/p>\r\n\r\n<div id=\"fs-id1558068\" class=\"ui-has-child-title\" data-type=\"note\"><header>\r\n<h2 class=\"os-title\" data-type=\"title\"><span class=\"os-title-label\" data-type=\"\">Imaginary and Complex Numbers<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<p id=\"fs-id1689292\">A <span id=\"term-00002\" data-type=\"term\">complex number<\/span> is a number of the form<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>where\r\n<ul id=\"fs-id1531123\">\r\n \t<li><math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>a<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>a<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>is the real part of the complex number.<\/li>\r\n \t<li><math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>b<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>b<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>is the imaginary part of the complex number.<\/li>\r\n<\/ul>\r\n<p id=\"fs-id1402821\">If<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>b<\/mi><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>b<\/mi><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>then\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>is a real number. If\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>a<\/mi><mo>=<\/mo><mn>0<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>a<\/mi><mo>=<\/mo><mn>0<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>b<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>b<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>is not equal to 0, the complex number is called a pure imaginary number. An <span id=\"term-00003\" data-type=\"term\">imaginary number<\/span> is an even root of a negative number.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1266532\" class=\"how-to-notitle ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"How To\"><header>\r\n<h2 class=\"os-title\" data-type=\"title\" data-label-parent=\"How To\"><span class=\"os-title-label\">How To<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<p id=\"fs-id1578095\"><strong>Given an imaginary number, express it in the standard form of a complex number.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1269513\" type=\"1\">\r\n \t<li>Write\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mo>\u2212<\/mo><mi>a<\/mi>\r\n<\/mrow>\r\n<\/msqrt>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msqrt>\r\n<mrow>\r\n<mo>\u2212<\/mo><mi>a<\/mi>\r\n<\/mrow>\r\n<\/msqrt>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>as\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msqrt>\r\n<mi>a<\/mi>\r\n<\/msqrt>\r\n<msqrt>\r\n<mrow>\r\n<mn>\u22121<\/mn>\r\n<\/mrow>\r\n<\/msqrt>\r\n<mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msqrt>\r\n<mi>a<\/mi>\r\n<\/msqrt>\r\n<msqrt>\r\n<mrow>\r\n<mn>\u22121<\/mn>\r\n<\/mrow>\r\n<\/msqrt>\r\n<mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\r\n \t<li>Express\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>\u22121<\/mn>\r\n<\/mrow>\r\n<\/msqrt>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>\u22121<\/mn>\r\n<\/mrow>\r\n<\/msqrt>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>as\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>i<\/mi><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>i<\/mi><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\r\n \t<li>Write\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msqrt>\r\n<mi>a<\/mi>\r\n<\/msqrt>\r\n<mo>\u22c5<\/mo><mi>i<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msqrt>\r\n<mi>a<\/mi>\r\n<\/msqrt>\r\n<mo>\u22c5<\/mo><mi>i<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>in simplest form.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"Example_02_04_01\" class=\"ui-has-child-title\" data-type=\"example\"><header>\r\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span>\r\n<span class=\"os-number\">1<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"body\">\r\n<div id=\"fs-id1884617\" class=\"unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1228287\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<h3 data-type=\"title\">Expressing an Imaginary Number in Standard Form<\/h3>\r\n<p id=\"fs-id1274827\">Express<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>\u22129<\/mn>\r\n<\/mrow>\r\n<\/msqrt>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>\u22129<\/mn>\r\n<\/mrow>\r\n<\/msqrt>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>in standard form.\r\n\r\n<\/div>\r\n<\/div>\r\n<details id=\"fs-id1813876\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\"><summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary><section class=\"ui-body\" role=\"alert\">\r\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\r\n<div class=\"os-solution-container\">\r\n<div id=\"fs-id2443487\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><msqrt><mrow><mn>\u22129<\/mn><\/mrow><\/msqrt><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<msqrt>\r\n<mn>9<\/mn>\r\n<\/msqrt>\r\n<msqrt>\r\n<mrow>\r\n<mn>\u22121<\/mn><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mn>3<\/mn><mi>i<\/mi><\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><msqrt><mrow><mn>\u22129<\/mn><\/mrow><\/msqrt><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<msqrt>\r\n<mn>9<\/mn>\r\n<\/msqrt>\r\n<msqrt>\r\n<mrow>\r\n<mn>\u22121<\/mn><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mn>3<\/mn><mi>i<\/mi><\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id881430\">In standard form, this is<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>0<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>0<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/details><\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2375806\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\"><header>\r\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span>\r\n<span class=\"os-number\">#1<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<div id=\"ti_02_04_01\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1294134\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id3048344\">Express<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>\u221224<\/mn>\r\n<\/mrow>\r\n<\/msqrt>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>\u221224<\/mn>\r\n<\/mrow>\r\n<\/msqrt>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>in standard form.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1591834\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Plotting a Complex Number on the Complex Plane<\/h2>\r\n<p id=\"fs-id3176103\">We cannot plot complex numbers on a number line as we might real numbers. However, we can still represent them graphically. To represent a complex number, we need to address the two components of the number. We use the <span id=\"term-00004\" data-type=\"term\">complex plane<\/span>, which is a coordinate system in which the horizontal axis represents the real component and the vertical axis represents the imaginary component. Complex numbers are the points on the plane, expressed as ordered pairs<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>,<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>,<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>where\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>a<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>a<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>represents the coordinate for the horizontal axis and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>b<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>b<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>represents the coordinate for the vertical axis.\r\n<p id=\"fs-id2517292\">Let\u2019s consider the number<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>\u22122<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>\u22122<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>The real part of the complex number is\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>\u22122<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>\u22122<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>and the imaginary part is 3. We plot the ordered pair\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>\u22122<\/mn><mo>,<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>\u22122<\/mn><mo>,<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>to represent the complex number\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>\u22122<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>\u22122<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>as shown in <a class=\"autogenerated-content\" href=\"2-4-complex-numbers#Figure_02_04_003\">Figure 2<\/a><strong>.<\/strong>\r\n<div id=\"Figure_02_04_003\" class=\"os-figure\">\r\n<figure class=\"small\" data-id=\"Figure_02_04_003\"><span id=\"fs-id1245266\" data-type=\"media\" data-alt=\"Coordinate plane with the x and y axes ranging from negative 5 to 5. The point negative 2 plus 3i is plotted on the graph. An arrow extends leftward from the origin two units and then an arrow extends upward three units from the end of the previous arrow.\">\r\n<img src=\"https:\/\/openstax.org\/books\/college-algebra-2e\/pages\/src#fixme\" alt=\"Coordinate plane with the x and y axes ranging from negative 5 to 5. The point negative 2 plus 3i is plotted on the graph. An arrow extends leftward from the origin two units and then an arrow extends upward three units from the end of the previous arrow.\" width=\"357\" height=\"372\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/4bc24ed9c0f41a9c1eb3faf6eea40b42241d503b\" \/>\r\n<\/span><\/figure>\r\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Figure <\/span>\r\n<span class=\"os-number\">2<\/span><\/div>\r\n<\/div>\r\n<div id=\"fs-id1184889\" class=\"ui-has-child-title\" data-type=\"note\"><header>\r\n<h2 class=\"os-title\" data-type=\"title\"><span class=\"os-title-label\" data-type=\"\">Complex Plane<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<p id=\"fs-id3269164\">In the complex plane, the horizontal axis is the real axis, and the vertical axis is the imaginary axis, as shown in <a class=\"autogenerated-content\" href=\"2-4-complex-numbers#Figure_02_04_004\">Figure 3<\/a>.<\/p>\r\n\r\n<div id=\"Figure_02_04_004\" class=\"os-figure\">\r\n<figure class=\"small\" data-id=\"Figure_02_04_004\"><span id=\"fs-id1943089\" data-type=\"media\" data-alt=\"A blank coordinate plane with the x-axis labeled: real and the y-axis labeled: imaginary.\">\r\n<img src=\"https:\/\/openstax.org\/books\/college-algebra-2e\/pages\/src#fixme\" alt=\"A blank coordinate plane with the x-axis labeled: real and the y-axis labeled: imaginary.\" width=\"229\" height=\"208\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/f597ea5f207951735a0de498e02a07e8cd28e3f5\" \/>\r\n<\/span><\/figure>\r\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Figure <\/span>\r\n<span class=\"os-number\">3<\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1273812\" class=\"how-to-notitle ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"How To\"><header>\r\n<h2 class=\"os-title\" data-type=\"title\" data-label-parent=\"How To\"><span class=\"os-title-label\">How To<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<p id=\"fs-id1887259\"><strong>Given a complex number, represent its components on the complex plane.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id2484938\" type=\"1\">\r\n \t<li>Determine the real part and the imaginary part of the complex number.<\/li>\r\n \t<li>Move along the horizontal axis to show the real part of the number.<\/li>\r\n \t<li>Move parallel to the vertical axis to show the imaginary part of the number.<\/li>\r\n \t<li>Plot the point.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"Example_02_04_02\" class=\"ui-has-child-title\" data-type=\"example\"><header>\r\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span>\r\n<span class=\"os-number\">2<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"body\">\r\n<div id=\"fs-id3113300\" class=\"unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1787353\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<h3 data-type=\"title\">Plotting a Complex Number on the Complex Plane<\/h3>\r\n<p id=\"fs-id1516111\">Plot the complex number<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>3<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi><mspace width=\"0.5em\"><\/mspace><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>3<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi><mspace width=\"0.5em\"><\/mspace><\/mrow><\/annotation-xml><\/semantics><\/math>on the complex plane.\r\n\r\n<\/div>\r\n<\/div>\r\n<details id=\"fs-id1148988\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\"><summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary><section class=\"ui-body\" role=\"alert\">\r\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\r\n<div class=\"os-solution-container\">\r\n<p id=\"fs-id2438175\">The real part of the complex number is<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>3<\/mn><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>3<\/mn><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>and the imaginary part is \u20134. We plot the ordered pair\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mo>,<\/mo><mn>\u22124<\/mn><mo stretchy=\"false\">)<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mo>,<\/mo><mn>\u22124<\/mn><mo stretchy=\"false\">)<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>as shown in <a class=\"autogenerated-content\" href=\"2-4-complex-numbers#Figure_02_04_005\">Figure 4<\/a><strong>.<\/strong>\r\n<div id=\"Figure_02_04_005\" class=\"os-figure\">\r\n<figure class=\"small\" data-id=\"Figure_02_04_005\"><span id=\"fs-id915324\" data-type=\"media\" data-alt=\"Coordinate plane with the x and y axes ranging from -5 to 5. The point 3 \u2013 4i is plotted, with an arrow extending rightward from the origin 3 units and an arrow extending downward 4 units from the end of the previous arrow.\">\r\n<img src=\"https:\/\/openstax.org\/books\/college-algebra-2e\/pages\/src#fixme\" alt=\"Coordinate plane with the x and y axes ranging from -5 to 5. The point 3 \u2013 4i is plotted, with an arrow extending rightward from the origin 3 units and an arrow extending downward 4 units from the end of the previous arrow.\" width=\"357\" height=\"372\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/5cad9c29f707a8f3cd1716c1288a2ca1eb8963aa\" \/>\r\n<\/span><\/figure>\r\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Figure <\/span>\r\n<span class=\"os-number\">4<\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/details><\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2512646\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\"><header>\r\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span>\r\n<span class=\"os-number\">#2<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<div id=\"ti_02_04_02\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1593307\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2507797\">Plot the complex number<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>\u22124<\/mn><mo>\u2212<\/mo><mi>i<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>\u22124<\/mn><mo>\u2212<\/mo><mi>i<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>on the complex plane.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1424098\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Adding and Subtracting Complex Numbers<\/h2>\r\n<p id=\"fs-id1463014\">Just as with real numbers, we can perform arithmetic operations on complex numbers. To add or subtract complex numbers, we combine the real parts and then combine the imaginary parts.<\/p>\r\n\r\n<div id=\"fs-id1442785\" class=\"ui-has-child-title\" data-type=\"note\"><header>\r\n<h2 class=\"os-title\" data-type=\"title\"><span class=\"os-title-label\" data-type=\"\">Complex Numbers: Addition and Subtraction<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<p id=\"fs-id1976732\">Adding complex numbers:<\/p>\r\n\r\n<div id=\"fs-id1519513\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>+<\/mo><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>c<\/mi><mo>+<\/mo><mi>d<\/mi><mi>i<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>=<\/mo><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>a<\/mi><mo>+<\/mo><mi>c<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>+<\/mo><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>b<\/mi><mo>+<\/mo><mi>d<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mi>i<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>+<\/mo><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>c<\/mi><mo>+<\/mo><mi>d<\/mi><mi>i<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>=<\/mo><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>a<\/mi><mo>+<\/mo><mi>c<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>+<\/mo><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>b<\/mi><mo>+<\/mo><mi>d<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mi>i<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1446903\">Subtracting complex numbers:<\/p>\r\n\r\n<div id=\"fs-id1018786\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>\u2212<\/mo><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>c<\/mi><mo>+<\/mo><mi>d<\/mi><mi>i<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>=<\/mo><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>a<\/mi><mo>\u2212<\/mo><mi>c<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>+<\/mo><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>b<\/mi><mo>\u2212<\/mo><mi>d<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mi>i<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>\u2212<\/mo><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>c<\/mi><mo>+<\/mo><mi>d<\/mi><mi>i<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>=<\/mo><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>a<\/mi><mo>\u2212<\/mo><mi>c<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>+<\/mo><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>b<\/mi><mo>\u2212<\/mo><mi>d<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mi>i<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1296613\" class=\"how-to-notitle ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"How To\"><header>\r\n<h2 class=\"os-title\" data-type=\"title\" data-label-parent=\"How To\"><span class=\"os-title-label\">How To<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<p id=\"fs-id2383706\"><strong>Given two complex numbers, find the sum or difference.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1805520\" type=\"1\">\r\n \t<li>Identify the real and imaginary parts of each number.<\/li>\r\n \t<li>Add or subtract the real parts.<\/li>\r\n \t<li>Add or subtract the imaginary parts.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"Example_02_04_03\" class=\"ui-has-child-title\" data-type=\"example\"><header>\r\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span>\r\n<span class=\"os-number\">3<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"body\">\r\n<div id=\"fs-id1133706\" class=\"unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1267465\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<h3 data-type=\"title\">Adding and Subtracting Complex Numbers<\/h3>\r\n<p id=\"fs-id1567983\">Add or subtract as indicated.<\/p>\r\n\r\n<ol id=\"fs-id3257388\" class=\"circled\" type=\"1\">\r\n \t<li><span class=\"token\">\u24d0<\/span>\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>3<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>+<\/mo><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>2<\/mn><mo>+<\/mo><mn>5<\/mn><mi>i<\/mi><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>3<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>+<\/mo><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>2<\/mn><mo>+<\/mo><mn>5<\/mn><mi>i<\/mi><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\r\n \t<li><span class=\"token\">\u24d1<\/span>\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22125<\/mn><mo>+<\/mo><mn>7<\/mn><mi>i<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>\u2212<\/mo><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u221211<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22125<\/mn><mo>+<\/mo><mn>7<\/mn><mi>i<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>\u2212<\/mo><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u221211<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<details id=\"fs-id1429773\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\"><summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary><section class=\"ui-body\" role=\"alert\">\r\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\r\n<div class=\"os-solution-container\">\r\n<p id=\"fs-id1288146\">We add the real parts and add the imaginary parts.<\/p>\r\n\r\n<ol id=\"fs-id2710031\" class=\"circled\" type=\"1\">\r\n \t<li><span class=\"token\">\u24d0<\/span><span data-type=\"newline\">\r\n<\/span>\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo>+<\/mo><mn>5<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow>\r\n<mn>3<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi><mo>+<\/mo><mn>2<\/mn><mo>+<\/mo><mn>5<\/mn><mi>i<\/mi><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mn>3<\/mn><mo>+<\/mo><mn>2<\/mn><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u22124<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>5<\/mn><mi>i<\/mi><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mo>+<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u22124<\/mn><mo>+<\/mo><mn>5<\/mn><mo stretchy=\"false\">)<\/mo><mi>i<\/mi><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>5<\/mn><mo>+<\/mo><mi>i<\/mi><\/mrow><\/mtd>\r\n<\/mtr>\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo>+<\/mo><mn>5<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow>\r\n<mn>3<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi><mo>+<\/mo><mn>2<\/mn><mo>+<\/mo><mn>5<\/mn><mi>i<\/mi><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mn>3<\/mn><mo>+<\/mo><mn>2<\/mn><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u22124<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>5<\/mn><mi>i<\/mi><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mo>+<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u22124<\/mn><mo>+<\/mo><mn>5<\/mn><mo stretchy=\"false\">)<\/mo><mi>i<\/mi><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>5<\/mn><mo>+<\/mo><mi>i<\/mi><\/mrow><\/mtd>\r\n<\/mtr>\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\r\n \t<li><span class=\"token\">\u24d1<\/span><span data-type=\"newline\">\r\n<\/span>\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>\u22125<\/mn><mo>+<\/mo><mn>7<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u221211<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>\u22125<\/mn><mo>+<\/mo><mn>7<\/mn><mi>i<\/mi><mo>+<\/mo><mn>11<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>i<\/mi><\/mrow><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>\u22125<\/mn><mo>+<\/mo><mn>11<\/mn><mo>+<\/mo><mn>7<\/mn><mi>i<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mi>i<\/mi><\/mrow><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mo stretchy=\"false\">(<\/mo><mn>\u22125<\/mn><mo>+<\/mo><mn>11<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mn>7<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><mi>i<\/mi><\/mrow><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>6<\/mn><mo>+<\/mo><mn>5<\/mn><mi>i<\/mi><\/mrow><\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>\u22125<\/mn><mo>+<\/mo><mn>7<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u221211<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>\u22125<\/mn><mo>+<\/mo><mn>7<\/mn><mi>i<\/mi><mo>+<\/mo><mn>11<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>i<\/mi><\/mrow><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>\u22125<\/mn><mo>+<\/mo><mn>11<\/mn><mo>+<\/mo><mn>7<\/mn><mi>i<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mi>i<\/mi><\/mrow><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mo stretchy=\"false\">(<\/mo><mn>\u22125<\/mn><mo>+<\/mo><mn>11<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mn>7<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><mi>i<\/mi><\/mrow><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>6<\/mn><mo>+<\/mo><mn>5<\/mn><mi>i<\/mi><\/mrow><\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/section><\/details><\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2302121\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\"><header>\r\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span>\r\n<span class=\"os-number\">#3<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<div id=\"ti_02_04_03\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1339021\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1334125\">Subtract<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>2<\/mn><mo>+<\/mo><mn>5<\/mn><mi>i<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>2<\/mn><mo>+<\/mo><mn>5<\/mn><mi>i<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>from\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>3<\/mn><mo>\u2013<\/mo><mn>4<\/mn><mi>i<\/mi><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>3<\/mn><mo>\u2013<\/mo><mn>4<\/mn><mi>i<\/mi><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1322676\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Multiplying Complex Numbers<\/h2>\r\n<p id=\"fs-id1188861\">Multiplying complex numbers is much like multiplying binomials. The major difference is that we work with the real and imaginary parts separately.<\/p>\r\n\r\n<section id=\"fs-id2995792\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Multiplying a Complex Number by a Real Number<\/h3>\r\n<p id=\"fs-id1533933\">Lets begin by multiplying a complex number by a real number. We distribute the real number just as we would with a binomial. Consider, for example,<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>3<\/mn><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>6<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>3<\/mn><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>6<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>:\r\n\r\n<span id=\"fs-id1545734\" data-type=\"media\" data-alt=\"Multiplication of a real number and a complex number. The 3 outside of the parentheses has arrows extending from it to both the 6 and the 2i inside of the parentheses. This expression is set equal to the quantity three times six plus the quantity three times two times i; this is the distributive property. The next line equals eighteen plus six times i; the simplification.\">\r\n<img src=\"https:\/\/openstax.org\/books\/college-algebra-2e\/pages\/src#fixme\" alt=\"Multiplication of a real number and a complex number. The 3 outside of the parentheses has arrows extending from it to both the 6 and the 2i inside of the parentheses. This expression is set equal to the quantity three times six plus the quantity three times two times i; this is the distributive property. The next line equals eighteen plus six times i; the simplification.\" width=\"487\" height=\"87\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/cc0c0bc49d1474d8c158cb4600642d565a51fa2b\" \/>\r\n<\/span>\r\n<div id=\"fs-id1494245\" class=\"how-to-notitle ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"How To\"><header>\r\n<h2 class=\"os-title\" data-type=\"title\" data-label-parent=\"How To\"><span class=\"os-title-label\">How To<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<p id=\"fs-id2714950\"><strong>Given a complex number and a real number, multiply to find the product.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1170540\" type=\"1\">\r\n \t<li>Use the distributive property.<\/li>\r\n \t<li>Simplify.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"Example_02_04_04\" class=\"ui-has-child-title\" data-type=\"example\"><header>\r\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span>\r\n<span class=\"os-number\">4<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"body\">\r\n<div id=\"fs-id1203991\" class=\"unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1930439\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<h3 data-type=\"title\">Multiplying a Complex Number by a Real Number<\/h3>\r\n<p id=\"fs-id1846156\">Find the product<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>4<\/mn><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo>+<\/mo><mn>5<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>4<\/mn><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo>+<\/mo><mn>5<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<details id=\"fs-id2664861\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\"><summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary><section class=\"ui-body\" role=\"alert\">\r\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\r\n<div class=\"os-solution-container\">\r\n<p id=\"fs-id1509986\">Distribute the 4.<\/p>\r\n\r\n<div id=\"fs-id1133872\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mn>4<\/mn><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo>+<\/mo><mn>5<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo>\u22c5<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo>\u22c5<\/mo><mn>5<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>8<\/mn><mo>+<\/mo><mn>20<\/mn><mi>i<\/mi><\/mrow><\/mtd>\r\n<\/mtr>\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mn>4<\/mn><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo>+<\/mo><mn>5<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo>\u22c5<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo>\u22c5<\/mo><mn>5<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>8<\/mn><mo>+<\/mo><mn>20<\/mn><mi>i<\/mi><\/mrow><\/mtd>\r\n<\/mtr>\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<\/div>\r\n<\/section><\/details><\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1745333\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\"><header>\r\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span>\r\n<span class=\"os-number\">#4<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<div id=\"ti_02_04_04\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1211448\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2497768\">Find the product:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>5<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>i<\/mi><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>5<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>i<\/mi><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1342702\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Multiplying Complex Numbers Together<\/h3>\r\n<p id=\"fs-id2700341\">Now, let\u2019s multiply two complex numbers. We can use either the distributive property or more specifically the FOIL method because we are dealing with binomials. Recall that FOIL is an acronym for multiplying First, Inner, Outer, and Last terms together. The difference with complex numbers is that when we get a squared term,<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>it equals\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>\u22121.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>\u22121.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n<div id=\"fs-id832128\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>c<\/mi><mo>+<\/mo><mi>d<\/mi><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mi>a<\/mi><mi>c<\/mi><mo>+<\/mo><mi>a<\/mi><mi>d<\/mi><mi>i<\/mi><mo>+<\/mo><mi>b<\/mi><mi>c<\/mi><mi>i<\/mi><mo>+<\/mo><mi>b<\/mi><mi>d<\/mi><msup>\r\n<mi>i<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mi>a<\/mi><mi>c<\/mi><mo>+<\/mo><mi>a<\/mi><mi>d<\/mi><mi>i<\/mi><mo>+<\/mo><mi>b<\/mi><mi>c<\/mi><mi>i<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>d<\/mi><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow><mspace width=\"2em\"><\/mspace>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><mn>\u22121<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mi>c<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>d<\/mi><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mi>d<\/mi><mo>+<\/mo><mi>b<\/mi><mi>c<\/mi><mo stretchy=\"false\">)<\/mo><mi>i<\/mi><\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>Group\u00a0real\u00a0terms\u00a0and\u00a0imaginary\u00a0terms<\/mtext><mo>.<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>c<\/mi><mo>+<\/mo><mi>d<\/mi><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mi>a<\/mi><mi>c<\/mi><mo>+<\/mo><mi>a<\/mi><mi>d<\/mi><mi>i<\/mi><mo>+<\/mo><mi>b<\/mi><mi>c<\/mi><mi>i<\/mi><mo>+<\/mo><mi>b<\/mi><mi>d<\/mi><msup>\r\n<mi>i<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mi>a<\/mi><mi>c<\/mi><mo>+<\/mo><mi>a<\/mi><mi>d<\/mi><mi>i<\/mi><mo>+<\/mo><mi>b<\/mi><mi>c<\/mi><mi>i<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>d<\/mi><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow><mspace width=\"2em\"><\/mspace>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><mn>\u22121<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mi>c<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>d<\/mi><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mi>d<\/mi><mo>+<\/mo><mi>b<\/mi><mi>c<\/mi><mo stretchy=\"false\">)<\/mo><mi>i<\/mi><\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>Group\u00a0real\u00a0terms\u00a0and\u00a0imaginary\u00a0terms<\/mtext><mo>.<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<div id=\"fs-id1758537\" class=\"how-to-notitle ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"How To\"><header>\r\n<h2 class=\"os-title\" data-type=\"title\" data-label-parent=\"How To\"><span class=\"os-title-label\">How To<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<p id=\"fs-id1457355\"><strong>Given two complex numbers, multiply to find the product.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1973809\" type=\"1\">\r\n \t<li>Use the distributive property or the FOIL method.<\/li>\r\n \t<li>Remember that\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><mn>\u22121.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><mn>\u22121.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\r\n \t<li>Group together the real terms and the imaginary terms<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"Example_02_04_05\" class=\"ui-has-child-title\" data-type=\"example\"><header>\r\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span>\r\n<span class=\"os-number\">5<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"body\">\r\n<div id=\"fs-id1311930\" class=\"unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id988839\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<h3 data-type=\"title\">Multiplying a Complex Number by a Complex Number<\/h3>\r\n<p id=\"fs-id1493312\">Multiply:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>4<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo>\u2212<\/mo><mn>5<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>4<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo>\u2212<\/mo><mn>5<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<details id=\"fs-id1435934\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\"><summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary><section class=\"ui-body\" role=\"alert\">\r\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\r\n<div class=\"os-solution-container\">\r\n<div id=\"fs-id1813679\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo>\u2212<\/mo><mn>5<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>4<\/mn><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mn>4<\/mn><mo stretchy=\"false\">(<\/mo><mn>5<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mn>5<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mn>8<\/mn><mo>\u2212<\/mo><mn>20<\/mn><mi>i<\/mi><mo>+<\/mo><mn>6<\/mn><mi>i<\/mi><mo>\u2212<\/mo><mn>15<\/mn><mo>(<\/mo><msup>\r\n<mi>i<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mo stretchy=\"false\">(<\/mo><mn>8<\/mn><mo>+<\/mo><mn>15<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u221220<\/mn><mo>+<\/mo><mn>6<\/mn><mo stretchy=\"false\">)<\/mo><mi>i<\/mi><\/mrow><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>23<\/mn><mo>\u2212<\/mo><mn>14<\/mn><mi>i<\/mi><\/mrow><\/mtd>\r\n<\/mtr>\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo>\u2212<\/mo><mn>5<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>4<\/mn><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mn>4<\/mn><mo stretchy=\"false\">(<\/mo><mn>5<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mn>5<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mn>8<\/mn><mo>\u2212<\/mo><mn>20<\/mn><mi>i<\/mi><mo>+<\/mo><mn>6<\/mn><mi>i<\/mi><mo>\u2212<\/mo><mn>15<\/mn><mo>(<\/mo><msup>\r\n<mi>i<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mo stretchy=\"false\">(<\/mo><mn>8<\/mn><mo>+<\/mo><mn>15<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u221220<\/mn><mo>+<\/mo><mn>6<\/mn><mo stretchy=\"false\">)<\/mo><mi>i<\/mi><\/mrow><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>23<\/mn><mo>\u2212<\/mo><mn>14<\/mn><mi>i<\/mi><\/mrow><\/mtd>\r\n<\/mtr>\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<\/div>\r\n<\/section><\/details><\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2486744\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\"><header>\r\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span>\r\n<span class=\"os-number\">#5<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<div id=\"ti_02_04_05\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2507020\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1475550\">Multiply:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>3<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi><\/mrow>\r\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>2<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>3<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi><\/mrow>\r\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>2<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/section><section id=\"fs-id1555648\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Dividing Complex Numbers<\/h2>\r\n<p id=\"fs-id1923471\">Dividing two complex numbers is more complicated than adding, subtracting, or multiplying because we cannot divide by an imaginary number, meaning that any fraction must have a real-number denominator to write the answer in standard form<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>We need to find a term by which we can multiply the numerator and the denominator that will eliminate the imaginary portion of the denominator so that we end up with a real number as the denominator. This term is called the complex conjugate of the denominator, which is found by changing the sign of the imaginary part of the complex number. In other words, the complex conjugate of\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>is\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>a<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>i<\/mi><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>a<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>i<\/mi><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>For example, the product of\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>a<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>i<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>a<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>i<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>is\r\n<div id=\"fs-id781562\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mrow>\r\n<msup>\r\n<mi>a<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mi>a<\/mi><mi>b<\/mi><mi>i<\/mi><mo>+<\/mo><mi>a<\/mi><mi>b<\/mi><mi>i<\/mi><mo>\u2212<\/mo><msup>\r\n<mi>b<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<msup>\r\n<mi>a<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><msup>\r\n<mi>b<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mrow>\r\n<msup>\r\n<mi>a<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mi>a<\/mi><mi>b<\/mi><mi>i<\/mi><mo>+<\/mo><mi>a<\/mi><mi>b<\/mi><mi>i<\/mi><mo>\u2212<\/mo><msup>\r\n<mi>b<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<msup>\r\n<mi>a<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><msup>\r\n<mi>b<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1965413\">The result is a real number.<\/p>\r\n<p id=\"fs-id1466166\">Note that complex conjugates have an opposite relationship: The complex conjugate of<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>is\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>a<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>i<\/mi><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>a<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>i<\/mi><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>and the complex conjugate of\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>a<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>i<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>a<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>i<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>is\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>Further, when a quadratic equation with real coefficients has complex solutions, the solutions are always complex conjugates of one another.\r\n<p id=\"fs-id2665321\">Suppose we want to divide<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>c<\/mi><mo>+<\/mo><mi>d<\/mi><mi>i<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>c<\/mi><mo>+<\/mo><mi>d<\/mi><mi>i<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>by\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>where neither\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>a<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>a<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>nor\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>b<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>b<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>equals zero. We first write the division as a fraction, then find the complex conjugate of the denominator, and multiply.\r\n<div id=\"fs-id1539267\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mi>c<\/mi><mo>+<\/mo><mi>d<\/mi><mi>i<\/mi>\r\n<\/mrow>\r\n<mrow>\r\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mspace width=\"0.5em\"><\/mspace><mtext>where\u00a0<\/mtext><mi>a<\/mi><mo>\u2260<\/mo><mn>0<\/mn><mspace width=\"0.5em\"><\/mspace><mtext>and\u00a0<\/mtext><mi>b<\/mi><mo>\u2260<\/mo><mn>0<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mrow>\r\n<mi>c<\/mi><mo>+<\/mo><mi>d<\/mi><mi>i<\/mi>\r\n<\/mrow>\r\n<mrow>\r\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mspace width=\"0.5em\"><\/mspace><mtext>where\u00a0<\/mtext><mi>a<\/mi><mo>\u2260<\/mo><mn>0<\/mn><mspace width=\"0.5em\"><\/mspace><mtext>and\u00a0<\/mtext><mi>b<\/mi><mo>\u2260<\/mo><mn>0<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id2504346\">Multiply the numerator and denominator by the complex conjugate of the denominator.<\/p>\r\n\r\n<div id=\"fs-id2015066\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>c<\/mi><mo>+<\/mo><mi>d<\/mi><mi>i<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>\u22c5<\/mo><mfrac>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>a<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>i<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>a<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>i<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>=<\/mo><mfrac>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>c<\/mi><mo>+<\/mo><mi>d<\/mi><mi>i<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>a<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>i<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>a<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>i<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>c<\/mi><mo>+<\/mo><mi>d<\/mi><mi>i<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>\u22c5<\/mo><mfrac>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>a<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>i<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>a<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>i<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>=<\/mo><mfrac>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>c<\/mi><mo>+<\/mo><mi>d<\/mi><mi>i<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>a<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>i<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>a<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>i<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id2498465\">Apply the distributive property.<\/p>\r\n\r\n<div id=\"fs-id2497874\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mo>=<\/mo><mfrac>\r\n<mrow>\r\n<mi>c<\/mi><mi>a<\/mi><mo>\u2212<\/mo><mi>c<\/mi><mi>b<\/mi><mi>i<\/mi><mo>+<\/mo><mi>a<\/mi><mi>d<\/mi><mi>i<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>d<\/mi><msup>\r\n<mi>i<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<mrow>\r\n<msup>\r\n<mi>a<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mi>a<\/mi><mi>b<\/mi><mi>i<\/mi><mo>+<\/mo><mi>a<\/mi><mi>b<\/mi><mi>i<\/mi><mo>\u2212<\/mo><msup>\r\n<mi>b<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mo>=<\/mo><mfrac>\r\n<mrow>\r\n<mi>c<\/mi><mi>a<\/mi><mo>\u2212<\/mo><mi>c<\/mi><mi>b<\/mi><mi>i<\/mi><mo>+<\/mo><mi>a<\/mi><mi>d<\/mi><mi>i<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>d<\/mi><msup>\r\n<mi>i<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<mrow>\r\n<msup>\r\n<mi>a<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mi>a<\/mi><mi>b<\/mi><mi>i<\/mi><mo>+<\/mo><mi>a<\/mi><mi>b<\/mi><mi>i<\/mi><mo>\u2212<\/mo><msup>\r\n<mi>b<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1790276\">Simplify, remembering that<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><mn>\u22121.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><mn>\u22121.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n<div id=\"fs-id2841772\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mo>=<\/mo><mfrac>\r\n<mrow>\r\n<mi>c<\/mi><mi>a<\/mi><mo>\u2212<\/mo><mi>c<\/mi><mi>b<\/mi><mi>i<\/mi><mo>+<\/mo><mi>a<\/mi><mi>d<\/mi><mi>i<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>d<\/mi><mo stretchy=\"false\">(<\/mo><mn>\u22121<\/mn><mo stretchy=\"false\">)<\/mo>\r\n<\/mrow>\r\n<mrow>\r\n<msup>\r\n<mi>a<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mi>a<\/mi><mi>b<\/mi><mi>i<\/mi><mo>+<\/mo><mi>a<\/mi><mi>b<\/mi><mi>i<\/mi><mo>\u2212<\/mo><msup>\r\n<mi>b<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo stretchy=\"false\">(<\/mo><mn>\u22121<\/mn><mo stretchy=\"false\">)<\/mo>\r\n<\/mrow>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mo>=<\/mo><mfrac>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mi>c<\/mi><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>d<\/mi><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mi>d<\/mi><mo>\u2212<\/mo><mi>c<\/mi><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><mi>i<\/mi>\r\n<\/mrow>\r\n<mrow>\r\n<msup>\r\n<mi>a<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><msup>\r\n<mi>b<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mo>=<\/mo><mfrac>\r\n<mrow>\r\n<mi>c<\/mi><mi>a<\/mi><mo>\u2212<\/mo><mi>c<\/mi><mi>b<\/mi><mi>i<\/mi><mo>+<\/mo><mi>a<\/mi><mi>d<\/mi><mi>i<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>d<\/mi><mo stretchy=\"false\">(<\/mo><mn>\u22121<\/mn><mo stretchy=\"false\">)<\/mo>\r\n<\/mrow>\r\n<mrow>\r\n<msup>\r\n<mi>a<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mi>a<\/mi><mi>b<\/mi><mi>i<\/mi><mo>+<\/mo><mi>a<\/mi><mi>b<\/mi><mi>i<\/mi><mo>\u2212<\/mo><msup>\r\n<mi>b<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo stretchy=\"false\">(<\/mo><mn>\u22121<\/mn><mo stretchy=\"false\">)<\/mo>\r\n<\/mrow>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mo>=<\/mo><mfrac>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mi>c<\/mi><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>d<\/mi><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mi>d<\/mi><mo>\u2212<\/mo><mi>c<\/mi><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><mi>i<\/mi>\r\n<\/mrow>\r\n<mrow>\r\n<msup>\r\n<mi>a<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><msup>\r\n<mi>b<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<div id=\"fs-id1392562\" class=\"ui-has-child-title\" data-type=\"note\"><header>\r\n<h2 class=\"os-title\" data-type=\"title\"><span class=\"os-title-label\" data-type=\"\">The Complex Conjugate<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<p id=\"fs-id1280042\">The <span id=\"term-00005\" data-type=\"term\">complex conjugate<\/span> of a complex number<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>is\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>a<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>i<\/mi><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>a<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>i<\/mi><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>It is found by changing the sign of the imaginary part of the complex number. The real part of the number is left unchanged.\r\n<ul id=\"fs-id1435532\">\r\n \t<li>When a complex number is multiplied by its complex conjugate, the result is a real number.<\/li>\r\n \t<li>When a complex number is added to its complex conjugate, the result is a real number.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"Example_02_04_06\" class=\"ui-has-child-title\" data-type=\"example\"><header>\r\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span>\r\n<span class=\"os-number\">6<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"body\">\r\n<div id=\"fs-id1332427\" class=\"unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1208193\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<h3 data-type=\"title\">Finding Complex Conjugates<\/h3>\r\n<p id=\"fs-id791546\">Find the complex conjugate of each number.<\/p>\r\n\r\n<ol id=\"fs-id1183222\" class=\"circled\" type=\"1\">\r\n \t<li><span class=\"token\">\u24d0<\/span>\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>2<\/mn><mo>+<\/mo><mi>i<\/mi><msqrt>\r\n<mn>5<\/mn>\r\n<\/msqrt>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>2<\/mn><mo>+<\/mo><mi>i<\/mi><msqrt>\r\n<mn>5<\/mn>\r\n<\/msqrt>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\r\n \t<li><span class=\"token\">\u24d1<\/span>\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mi>i<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mi>i<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<details id=\"fs-id2935900\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\"><summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary><section class=\"ui-body\" role=\"alert\">\r\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\r\n<div class=\"os-solution-container\">\r\n<ol id=\"fs-id1216223\" class=\"circled\" type=\"1\">\r\n \t<li><span class=\"token\">\u24d0<\/span>The number is already in the form\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>The complex conjugate is\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>a<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>i<\/mi><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>a<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>i<\/mi><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>or\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>2<\/mn><mo>\u2212<\/mo><mi>i<\/mi><msqrt>\r\n<mn>5<\/mn>\r\n<\/msqrt>\r\n<mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>2<\/mn><mo>\u2212<\/mo><mi>i<\/mi><msqrt>\r\n<mn>5<\/mn>\r\n<\/msqrt>\r\n<mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\r\n \t<li><span class=\"token\">\u24d1<\/span>We can rewrite this number in the form\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>as\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>0<\/mn><mo>\u2212<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mi>i<\/mi><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>0<\/mn><mo>\u2212<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mi>i<\/mi><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>The complex conjugate is\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>a<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>i<\/mi><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>a<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>i<\/mi><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>or\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>0<\/mn><mo>+<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mi>i<\/mi><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>0<\/mn><mo>+<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mi>i<\/mi><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>This can be written simply as\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mi>i<\/mi><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mi>i<\/mi><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/section><\/details>\r\n<div id=\"fs-id1425864\" data-type=\"commentary\">\r\n<h3 data-type=\"commentary-title\"><span class=\"os-title-label\">Analysis <\/span><\/h3>\r\n<p id=\"fs-id2315664\">Although we have seen that we can find the complex conjugate of an imaginary number, in practice we generally find the complex conjugates of only complex numbers with both a real and an imaginary component. To obtain a real number from an imaginary number, we can simply multiply by<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>i<\/mi><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>i<\/mi><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1445609\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\"><header>\r\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span>\r\n<span class=\"os-number\">#6<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<div id=\"ti_02_04_06\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1562514\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1562515\">Find the complex conjugate of<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>\u22123<\/mn><mo>+<\/mo><mn>4<\/mn><mi>i<\/mi><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>\u22123<\/mn><mo>+<\/mo><mn>4<\/mn><mi>i<\/mi><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2521760\" class=\"how-to-notitle ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"How To\"><header>\r\n<h2 class=\"os-title\" data-type=\"title\" data-label-parent=\"How To\"><span class=\"os-title-label\">How To<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<p id=\"fs-id1395747\"><strong>Given two complex numbers, divide one by the other.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id2753766\" type=\"1\">\r\n \t<li>Write the division problem as a fraction.<\/li>\r\n \t<li>Determine the complex conjugate of the denominator.<\/li>\r\n \t<li>Multiply the numerator and denominator of the fraction by the complex conjugate of the denominator.<\/li>\r\n \t<li>Simplify.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"Example_02_04_07\" class=\"ui-has-child-title\" data-type=\"example\"><header>\r\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span>\r\n<span class=\"os-number\">7<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"body\">\r\n<div id=\"fs-id1462292\" class=\"unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1883764\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<h3 data-type=\"title\">Dividing Complex Numbers<\/h3>\r\n<p id=\"fs-id1531602\">Divide:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>2<\/mn><mo>+<\/mo><mn>5<\/mn><mi>i<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>2<\/mn><mo>+<\/mo><mn>5<\/mn><mi>i<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>by\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>4<\/mn><mo>\u2212<\/mo><mi>i<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>4<\/mn><mo>\u2212<\/mo><mi>i<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<details id=\"fs-id1511138\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\"><summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary><section class=\"ui-body\" role=\"alert\">\r\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\r\n<div class=\"os-solution-container\">\r\n<p id=\"fs-id2483518\">We begin by writing the problem as a fraction.<\/p>\r\n\r\n<div id=\"fs-id1736666\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>2<\/mn><mo>+<\/mo><mn>5<\/mn><mi>i<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>4<\/mn><mo>\u2212<\/mo><mi>i<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>2<\/mn><mo>+<\/mo><mn>5<\/mn><mi>i<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>4<\/mn><mo>\u2212<\/mo><mi>i<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id3119235\">Then we multiply the numerator and denominator by the complex conjugate of the denominator.<\/p>\r\n\r\n<div id=\"fs-id1500527\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo>+<\/mo><mn>5<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo>\r\n<\/mrow>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo>\u2212<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mspace width=\"0.5em\"><\/mspace><mo>\u22c5<\/mo><mfrac>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo>+<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo>\r\n<\/mrow>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo>+<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo>\r\n<\/mrow>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo>+<\/mo><mn>5<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo>\r\n<\/mrow>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo>\u2212<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mspace width=\"0.5em\"><\/mspace><mo>\u22c5<\/mo><mfrac>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo>+<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo>\r\n<\/mrow>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo>+<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo>\r\n<\/mrow>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1204485\">To multiply two complex numbers, we expand the product as we would with polynomials (using FOIL).<\/p>\r\n\r\n<div id=\"fs-id2906059\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow>\r\n<mfrac>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo>+<\/mo><mn>5<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo>\u2212<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mfrac>\r\n<mo>\u22c5<\/mo><mfrac>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo>+<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo>+<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>8<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi><mo>+<\/mo><mn>20<\/mn><mi>i<\/mi><mo>+<\/mo><mn>5<\/mn><msup>\r\n<mi>i<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<\/mrow>\r\n<mrow>\r\n<mn>16<\/mn><mo>+<\/mo><mn>4<\/mn><mi>i<\/mi><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi><mo>\u2212<\/mo><msup>\r\n<mi>i<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<\/mrow>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>8<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi><mo>+<\/mo><mn>20<\/mn><mi>i<\/mi><mo>+<\/mo><mn>5<\/mn><mo stretchy=\"false\">(<\/mo><mn>\u22121<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<mrow>\r\n<mn>16<\/mn><mo>+<\/mo><mn>4<\/mn><mi>i<\/mi><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u22121<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>Because<\/mtext><mo>\u00a0<\/mo><msup>\r\n<mi>i<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><mn>\u22121.<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>3<\/mn><mo>+<\/mo><mn>22<\/mn><mi>i<\/mi><\/mrow>\r\n<mrow>\r\n<mn>17<\/mn><\/mrow>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mfrac>\r\n<mn>3<\/mn>\r\n<mrow>\r\n<mn>17<\/mn><\/mrow>\r\n<\/mfrac>\r\n<mo>+<\/mo><mfrac>\r\n<mrow>\r\n<mn>22<\/mn><\/mrow>\r\n<mrow>\r\n<mn>17<\/mn><\/mrow>\r\n<\/mfrac>\r\n<mi>i<\/mi><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>Separate\u00a0real\u00a0and\u00a0imaginary\u00a0parts<\/mtext><mo>.<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow>\r\n<mfrac>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo>+<\/mo><mn>5<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo>\u2212<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mfrac>\r\n<mo>\u22c5<\/mo><mfrac>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo>+<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo>+<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>8<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi><mo>+<\/mo><mn>20<\/mn><mi>i<\/mi><mo>+<\/mo><mn>5<\/mn><msup>\r\n<mi>i<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<\/mrow>\r\n<mrow>\r\n<mn>16<\/mn><mo>+<\/mo><mn>4<\/mn><mi>i<\/mi><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi><mo>\u2212<\/mo><msup>\r\n<mi>i<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<\/mrow>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>8<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi><mo>+<\/mo><mn>20<\/mn><mi>i<\/mi><mo>+<\/mo><mn>5<\/mn><mo stretchy=\"false\">(<\/mo><mn>\u22121<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<mrow>\r\n<mn>16<\/mn><mo>+<\/mo><mn>4<\/mn><mi>i<\/mi><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u22121<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>Because<\/mtext><mo>\u00a0<\/mo><msup>\r\n<mi>i<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><mn>\u22121.<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>3<\/mn><mo>+<\/mo><mn>22<\/mn><mi>i<\/mi><\/mrow>\r\n<mrow>\r\n<mn>17<\/mn><\/mrow>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mfrac>\r\n<mn>3<\/mn>\r\n<mrow>\r\n<mn>17<\/mn><\/mrow>\r\n<\/mfrac>\r\n<mo>+<\/mo><mfrac>\r\n<mrow>\r\n<mn>22<\/mn><\/mrow>\r\n<mrow>\r\n<mn>17<\/mn><\/mrow>\r\n<\/mfrac>\r\n<mi>i<\/mi><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>Separate\u00a0real\u00a0and\u00a0imaginary\u00a0parts<\/mtext><mo>.<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id2486676\">Note that this expresses the quotient in standard form.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/details><\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1555717\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Simplifying Powers of <em data-effect=\"italics\">i<\/em><\/h2>\r\n<p id=\"fs-id2049707\">The powers of<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>i<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>i<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>are cyclic. Let\u2019s look at what happens when we raise\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>i<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>i<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>to increasing powers.\r\n<div id=\"fs-id1537277\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mn>1<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><mi>i<\/mi>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><mn>\u22121<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mn>3<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><msup>\r\n<mi>i<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u22c5<\/mo><mi>i<\/mi><mo>=<\/mo><mn>\u22121<\/mn><mo>\u22c5<\/mo><mi>i<\/mi><mo>=<\/mo><mi>\u2212<\/mi><mi>i<\/mi>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mn>4<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><msup>\r\n<mi>i<\/mi>\r\n<mn>3<\/mn>\r\n<\/msup>\r\n<mo>\u22c5<\/mo><mi>i<\/mi><mo>=<\/mo><mi>\u2212<\/mi><mi>i<\/mi><mo>\u22c5<\/mo><mi>i<\/mi><mo>=<\/mo><mi>\u2212<\/mi><msup>\r\n<mi>i<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u22121<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mn>5<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><msup>\r\n<mi>i<\/mi>\r\n<mn>4<\/mn>\r\n<\/msup>\r\n<mo>\u22c5<\/mo><mi>i<\/mi><mo>=<\/mo><mn>1<\/mn><mo>\u22c5<\/mo><mi>i<\/mi><mo>=<\/mo><mi>i<\/mi>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mn>1<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><mi>i<\/mi>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><mn>\u22121<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mn>3<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><msup>\r\n<mi>i<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u22c5<\/mo><mi>i<\/mi><mo>=<\/mo><mn>\u22121<\/mn><mo>\u22c5<\/mo><mi>i<\/mi><mo>=<\/mo><mi>\u2212<\/mi><mi>i<\/mi>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mn>4<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><msup>\r\n<mi>i<\/mi>\r\n<mn>3<\/mn>\r\n<\/msup>\r\n<mo>\u22c5<\/mo><mi>i<\/mi><mo>=<\/mo><mi>\u2212<\/mi><mi>i<\/mi><mo>\u22c5<\/mo><mi>i<\/mi><mo>=<\/mo><mi>\u2212<\/mi><msup>\r\n<mi>i<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u22121<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mn>5<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><msup>\r\n<mi>i<\/mi>\r\n<mn>4<\/mn>\r\n<\/msup>\r\n<mo>\u22c5<\/mo><mi>i<\/mi><mo>=<\/mo><mn>1<\/mn><mo>\u22c5<\/mo><mi>i<\/mi><mo>=<\/mo><mi>i<\/mi>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id2498837\">We can see that when we get to the fifth power of<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>i<\/mi><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>i<\/mi><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>it is equal to the first power. As we continue to multiply\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>i<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>i<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>by increasing powers, we will see a cycle of four. Let\u2019s examine the next four powers of\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>i<\/mi><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>i<\/mi><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n<div id=\"fs-id1943372\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mn>6<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><msup>\r\n<mi>i<\/mi>\r\n<mn>5<\/mn>\r\n<\/msup>\r\n<mo>\u22c5<\/mo><mi>i<\/mi><mo>=<\/mo><mi>i<\/mi><mo>\u22c5<\/mo><mi>i<\/mi><mo>=<\/mo><msup>\r\n<mi>i<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><mn>\u22121<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mn>7<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><msup>\r\n<mi>i<\/mi>\r\n<mn>6<\/mn>\r\n<\/msup>\r\n<mo>\u22c5<\/mo><mi>i<\/mi><mo>=<\/mo><msup>\r\n<mi>i<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u22c5<\/mo><mi>i<\/mi><mo>=<\/mo><msup>\r\n<mi>i<\/mi>\r\n<mn>3<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><mi>\u2212<\/mi><mi>i<\/mi>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mn>8<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><msup>\r\n<mi>i<\/mi>\r\n<mn>7<\/mn>\r\n<\/msup>\r\n<mo>\u22c5<\/mo><mi>i<\/mi><mo>=<\/mo><msup>\r\n<mi>i<\/mi>\r\n<mn>3<\/mn>\r\n<\/msup>\r\n<mo>\u22c5<\/mo><mi>i<\/mi><mo>=<\/mo><msup>\r\n<mi>i<\/mi>\r\n<mn>4<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mn>9<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><msup>\r\n<mi>i<\/mi>\r\n<mn>8<\/mn>\r\n<\/msup>\r\n<mo>\u22c5<\/mo><mi>i<\/mi><mo>=<\/mo><msup>\r\n<mi>i<\/mi>\r\n<mn>4<\/mn>\r\n<\/msup>\r\n<mo>\u22c5<\/mo><mi>i<\/mi><mo>=<\/mo><msup>\r\n<mi>i<\/mi>\r\n<mn>5<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><mi>i<\/mi>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mn>6<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><msup>\r\n<mi>i<\/mi>\r\n<mn>5<\/mn>\r\n<\/msup>\r\n<mo>\u22c5<\/mo><mi>i<\/mi><mo>=<\/mo><mi>i<\/mi><mo>\u22c5<\/mo><mi>i<\/mi><mo>=<\/mo><msup>\r\n<mi>i<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><mn>\u22121<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mn>7<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><msup>\r\n<mi>i<\/mi>\r\n<mn>6<\/mn>\r\n<\/msup>\r\n<mo>\u22c5<\/mo><mi>i<\/mi><mo>=<\/mo><msup>\r\n<mi>i<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u22c5<\/mo><mi>i<\/mi><mo>=<\/mo><msup>\r\n<mi>i<\/mi>\r\n<mn>3<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><mi>\u2212<\/mi><mi>i<\/mi>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mn>8<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><msup>\r\n<mi>i<\/mi>\r\n<mn>7<\/mn>\r\n<\/msup>\r\n<mo>\u22c5<\/mo><mi>i<\/mi><mo>=<\/mo><msup>\r\n<mi>i<\/mi>\r\n<mn>3<\/mn>\r\n<\/msup>\r\n<mo>\u22c5<\/mo><mi>i<\/mi><mo>=<\/mo><msup>\r\n<mi>i<\/mi>\r\n<mn>4<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mn>9<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><msup>\r\n<mi>i<\/mi>\r\n<mn>8<\/mn>\r\n<\/msup>\r\n<mo>\u22c5<\/mo><mi>i<\/mi><mo>=<\/mo><msup>\r\n<mi>i<\/mi>\r\n<mn>4<\/mn>\r\n<\/msup>\r\n<mo>\u22c5<\/mo><mi>i<\/mi><mo>=<\/mo><msup>\r\n<mi>i<\/mi>\r\n<mn>5<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><mi>i<\/mi>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1442394\">The cycle is repeated continuously:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>i<\/mi><mo>,<\/mo><mn>\u22121<\/mn><mo>,<\/mo><mo>\u2212<\/mo><mi>i<\/mi><mo>,<\/mo><mn>1<\/mn><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>i<\/mi><mo>,<\/mo><mn>\u22121<\/mn><mo>,<\/mo><mo>\u2212<\/mo><mi>i<\/mi><mo>,<\/mo><mn>1<\/mn><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>every four powers.\r\n<div id=\"Example_02_04_08\" class=\"ui-has-child-title\" data-type=\"example\"><header>\r\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span>\r\n<span class=\"os-number\">8<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"body\">\r\n<div id=\"fs-id1831392\" class=\"unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2440261\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<h3 data-type=\"title\">Simplifying Powers of <math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>i<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>i<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/h3>\r\n<p id=\"fs-id1716914\">Evaluate:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mrow>\r\n<mn>35<\/mn>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mrow>\r\n<mn>35<\/mn>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<details id=\"fs-id2435624\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\"><summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary><section class=\"ui-body\" role=\"alert\">\r\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\r\n<div class=\"os-solution-container\">\r\n<p id=\"fs-id1269680\">Since<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mn>4<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><mn>1<\/mn><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mn>4<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><mn>1<\/mn><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>we can simplify the problem by factoring out as many factors of\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mn>4<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mn>4<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>as possible. To do so, first determine how many times 4 goes into 35:\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>35<\/mn><mo>=<\/mo><mn>4<\/mn><mo>\u22c5<\/mo><mn>8<\/mn><mo>+<\/mo><mn>3.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>35<\/mn><mo>=<\/mo><mn>4<\/mn><mo>\u22c5<\/mo><mn>8<\/mn><mo>+<\/mo><mn>3.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n<div id=\"fs-id1257595\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mrow>\r\n<mn>35<\/mn>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>=<\/mo><msup>\r\n<mi>i<\/mi>\r\n<mrow>\r\n<mn>4<\/mn><mo>\u22c5<\/mo><mn>8<\/mn><mo>+<\/mo><mn>3<\/mn>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>=<\/mo><msup>\r\n<mi>i<\/mi>\r\n<mrow>\r\n<mn>4<\/mn><mo>\u22c5<\/mo><mn>8<\/mn>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>\u22c5<\/mo><msup>\r\n<mi>i<\/mi>\r\n<mn>3<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mn>4<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<mn>8<\/mn>\r\n<\/msup>\r\n<mo>\u22c5<\/mo><msup>\r\n<mi>i<\/mi>\r\n<mn>3<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><msup>\r\n<mn>1<\/mn>\r\n<mn>8<\/mn>\r\n<\/msup>\r\n<mo>\u22c5<\/mo><msup>\r\n<mi>i<\/mi>\r\n<mn>3<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><msup>\r\n<mi>i<\/mi>\r\n<mn>3<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><mo>\u2212<\/mo><mi>i<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mrow>\r\n<mn>35<\/mn>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>=<\/mo><msup>\r\n<mi>i<\/mi>\r\n<mrow>\r\n<mn>4<\/mn><mo>\u22c5<\/mo><mn>8<\/mn><mo>+<\/mo><mn>3<\/mn>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>=<\/mo><msup>\r\n<mi>i<\/mi>\r\n<mrow>\r\n<mn>4<\/mn><mo>\u22c5<\/mo><mn>8<\/mn>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>\u22c5<\/mo><msup>\r\n<mi>i<\/mi>\r\n<mn>3<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mn>4<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<mn>8<\/mn>\r\n<\/msup>\r\n<mo>\u22c5<\/mo><msup>\r\n<mi>i<\/mi>\r\n<mn>3<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><msup>\r\n<mn>1<\/mn>\r\n<mn>8<\/mn>\r\n<\/msup>\r\n<mo>\u22c5<\/mo><msup>\r\n<mi>i<\/mi>\r\n<mn>3<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><msup>\r\n<mi>i<\/mi>\r\n<mn>3<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><mo>\u2212<\/mo><mi>i<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<\/div>\r\n<\/section><\/details><\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1845345\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\"><header>\r\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span>\r\n<span class=\"os-number\">#7<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<div id=\"ti_02_04_07\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1745095\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1745096\">Evaluate:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mrow>\r\n<mn>18<\/mn>\r\n<\/mrow>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mrow>\r\n<mn>18<\/mn>\r\n<\/mrow>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1567766\" class=\"precalculus qa ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Q&amp;A\"><header>\r\n<h2 class=\"os-title\" data-type=\"title\" data-label-parent=\"Q&amp;A\"><span class=\"os-title-label\">Q&amp;A<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<p id=\"fs-id2683377\"><strong>Can we write <\/strong><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mrow>\r\n<mn>35<\/mn>\r\n<\/mrow>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mrow>\r\n<mn>35<\/mn>\r\n<\/mrow>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>in other helpful ways?\r\n<p id=\"fs-id1557516\"><em data-effect=\"italics\">As we saw in <a class=\"autogenerated-content\" href=\"2-4-complex-numbers#Example_02_04_08\">Example 8<\/a>, we reduced <\/em><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mrow>\r\n<mn>35<\/mn>\r\n<\/mrow>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mrow>\r\n<mn>35<\/mn>\r\n<\/mrow>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>to\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mn>3<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mn>3<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>by dividing the exponent by 4 and using the remainder to find the simplified form. But perhaps another factorization of\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mrow>\r\n<mn>35<\/mn>\r\n<\/mrow>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mrow>\r\n<mn>35<\/mn>\r\n<\/mrow>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>may be more useful. <a class=\"autogenerated-content\" href=\"2-4-complex-numbers#Table_02_04_01\">Table 1<\/a> shows some other possible factorizations.\r\n<div id=\"Table_02_04_01\" class=\"os-table\">\r\n<table data-id=\"Table_02_04_01\">\r\n<tbody>\r\n<tr>\r\n<td data-align=\"center\"><strong>Factorization of<\/strong>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mrow>\r\n<mn>35<\/mn>\r\n<\/mrow>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mrow>\r\n<mn>35<\/mn>\r\n<\/mrow>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\r\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mrow>\r\n<mn>34<\/mn>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>\u22c5<\/mo><mi>i<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mrow>\r\n<mn>34<\/mn>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>\u22c5<\/mo><mi>i<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\r\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mrow>\r\n<mn>33<\/mn>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>\u22c5<\/mo><msup>\r\n<mi>i<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mrow>\r\n<mn>33<\/mn>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>\u22c5<\/mo><msup>\r\n<mi>i<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\r\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mrow>\r\n<mn>31<\/mn>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>\u22c5<\/mo><msup>\r\n<mi>i<\/mi>\r\n<mn>4<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mrow>\r\n<mn>31<\/mn>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>\u22c5<\/mo><msup>\r\n<mi>i<\/mi>\r\n<mn>4<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\r\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mrow>\r\n<mn>19<\/mn>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>\u22c5<\/mo><msup>\r\n<mi>i<\/mi>\r\n<mrow>\r\n<mn>16<\/mn>\r\n<\/mrow>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mrow>\r\n<mn>19<\/mn>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>\u22c5<\/mo><msup>\r\n<mi>i<\/mi>\r\n<mrow>\r\n<mn>16<\/mn>\r\n<\/mrow>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\"><strong>Reduced form<\/strong><\/td>\r\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<mrow>\r\n<mn>17<\/mn>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>\u22c5<\/mo><mi>i<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<mrow>\r\n<mn>17<\/mn>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>\u22c5<\/mo><mi>i<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\r\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mrow>\r\n<mn>33<\/mn>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>\u22c5<\/mo><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22121<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mrow>\r\n<mn>33<\/mn>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>\u22c5<\/mo><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22121<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\r\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mrow>\r\n<mn>31<\/mn>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>\u22c5<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mrow>\r\n<mn>31<\/mn>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>\u22c5<\/mo><mn>1<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\r\n<td data-align=\"center\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mrow>\r\n<mn>19<\/mn>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>\u22c5<\/mo><msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mn>4<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<mn>4<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mrow>\r\n<mn>19<\/mn>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>\u22c5<\/mo><msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mn>4<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<mn>4<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\"><strong>Simplified form<\/strong><\/td>\r\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22121<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<mrow>\r\n<mn>17<\/mn>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>\u22c5<\/mo><mi>i<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22121<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<mrow>\r\n<mn>17<\/mn>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>\u22c5<\/mo><mi>i<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\r\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mo>\u2212<\/mo><msup>\r\n<mi>i<\/mi>\r\n<mrow>\r\n<mn>33<\/mn>\r\n<\/mrow>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mo>\u2212<\/mo><msup>\r\n<mi>i<\/mi>\r\n<mrow>\r\n<mn>33<\/mn>\r\n<\/mrow>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\r\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mrow>\r\n<mn>31<\/mn>\r\n<\/mrow>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mrow>\r\n<mn>31<\/mn>\r\n<\/mrow>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\r\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mrow>\r\n<mn>19<\/mn>\r\n<\/mrow>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mrow>\r\n<mn>19<\/mn>\r\n<\/mrow>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Table <\/span>\r\n<span class=\"os-number\">1<\/span>\r\n\r\n<span class=\"os-caption\"><span class=\"os-caption\">\r\n<\/span><\/span>&nbsp;\r\n\r\n<span class=\"os-caption\">\r\n<\/span><\/div>\r\n<\/div>\r\n<p id=\"fs-id1164415\"><em data-effect=\"italics\">Each of these will eventually result in the answer we obtained above but may require several more steps than our earlier method.<\/em><\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2888800\" class=\"media-notitle ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Media\"><header>\r\n<h2 class=\"os-title\" data-type=\"title\" data-label-parent=\"Media\"><span class=\"os-title-label\">Media<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<p id=\"fs-id1339843\">Access these online resources for additional instruction and practice with complex numbers.<\/p>\r\n\r\n<ul id=\"fs-id2876017\">\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/addsubcomplex\" target=\"_blank\" rel=\"noopener nofollow\">Adding and Subtracting Complex Numbers<\/a><\/li>\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/multiplycomplex\" target=\"_blank\" rel=\"noopener nofollow\">Multiply Complex Numbers<\/a><\/li>\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/multcompconj\" target=\"_blank\" rel=\"noopener nofollow\">Multiplying Complex Conjugates<\/a><\/li>\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/raisingi\" target=\"_blank\" rel=\"noopener nofollow\">Raising <em data-effect=\"italics\">i<\/em> to Powers<\/a><\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section>\r\n<div class=\"os-eos os-section-exercises-container\" data-uuid-key=\".section-exercises\">\r\n<h2 data-type=\"document-title\" data-rex-keep=\"true\"><span class=\"os-text\">2.4 Section Exercises<\/span><\/h2>\r\n<section id=\"fs-id3236190\" class=\"section-exercises\" data-depth=\"1\"><section id=\"fs-id1399221\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Verbal<\/h3>\r\n<div id=\"fs-id1418488\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id767934\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1418488-solution\">1<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id767935\">Explain how to add complex numbers.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2638642\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1561233\" data-type=\"problem\"><span class=\"os-number\">2<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1561234\">What is the basic principle in multiplication of complex numbers?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1265940\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1265942\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1265940-solution\">3<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1181910\">Give an example to show that the product of two imaginary numbers is not always imaginary.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1289893\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1289894\" data-type=\"problem\"><span class=\"os-number\">4<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1289895\">What is a characteristic of the plot of a real number in the complex plane?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id2979908\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Algebraic<\/h3>\r\n<p id=\"fs-id1722892\">For the following exercises, evaluate the algebraic expressions.<\/p>\r\n\r\n<div id=\"fs-id1295468\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2906264\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1295468-solution\">5<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2906265\">If<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>y<\/mi><mo>=<\/mo><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><mo>,<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>y<\/mi><mo>=<\/mo><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><mo>,<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>evaluate\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>y<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>y<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>given\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>x<\/mi><mo>=<\/mo><mn>2<\/mn><mi>i<\/mi><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>x<\/mi><mo>=<\/mo><mn>2<\/mn><mi>i<\/mi><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1332284\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1332285\" data-type=\"problem\"><span class=\"os-number\">6<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id3229108\">If<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>y<\/mi><mo>=<\/mo><msup>\r\n<mi>x<\/mi>\r\n<mn>3<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>2<\/mn><mo>,<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>y<\/mi><mo>=<\/mo><msup>\r\n<mi>x<\/mi>\r\n<mn>3<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>2<\/mn><mo>,<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>evaluate\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>y<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>y<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>given\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>x<\/mi><mo>=<\/mo><mi>i<\/mi><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>x<\/mi><mo>=<\/mo><mi>i<\/mi><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1570762\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1570763\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1570762-solution\">7<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1463486\">If<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>y<\/mi><mo>=<\/mo><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>5<\/mn><mo>,<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>y<\/mi><mo>=<\/mo><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>5<\/mn><mo>,<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>evaluate\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>y<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>y<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>given\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>x<\/mi><mo>=<\/mo><mn>2<\/mn><mo>+<\/mo><mi>i<\/mi><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>x<\/mi><mo>=<\/mo><mn>2<\/mn><mo>+<\/mo><mi>i<\/mi><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1339584\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1339585\" data-type=\"problem\"><span class=\"os-number\">8<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1339586\">If<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>y<\/mi><mo>=<\/mo><mn>2<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo>,<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>y<\/mi><mo>=<\/mo><mn>2<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo>,<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>evaluate\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>y<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>y<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>given\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>x<\/mi><mo>=<\/mo><mn>2<\/mn><mo>\u2212<\/mo><mn>3<\/mn><mi>i<\/mi><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>x<\/mi><mo>=<\/mo><mn>2<\/mn><mo>\u2212<\/mo><mn>3<\/mn><mi>i<\/mi><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1701288\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1701289\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1701288-solution\">9<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1701290\">If<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>y<\/mi><mo>=<\/mo><mfrac>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<mrow>\r\n<mn>2<\/mn><mo>\u2212<\/mo><mi>x<\/mi>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>y<\/mi><mo>=<\/mo><mfrac>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<mrow>\r\n<mn>2<\/mn><mo>\u2212<\/mo><mi>x<\/mi>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>evaluate\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>y<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>y<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>given\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>x<\/mi><mo>=<\/mo><mn>5<\/mn><mi>i<\/mi><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>x<\/mi><mo>=<\/mo><mn>5<\/mn><mi>i<\/mi><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1353461\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id3008607\" data-type=\"problem\"><span class=\"os-number\">10<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id3008608\">If<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>y<\/mi><mo>=<\/mo><mfrac>\r\n<mrow>\r\n<mn>1<\/mn><mo>+<\/mo><mn>2<\/mn><mi>x<\/mi>\r\n<\/mrow>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>3<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>y<\/mi><mo>=<\/mo><mfrac>\r\n<mrow>\r\n<mn>1<\/mn><mo>+<\/mo><mn>2<\/mn><mi>x<\/mi>\r\n<\/mrow>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>3<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>evaluate\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>y<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>y<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>given\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>x<\/mi><mo>=<\/mo><mn>4<\/mn><mi>i<\/mi><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>x<\/mi><mo>=<\/mo><mn>4<\/mn><mi>i<\/mi><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1512803\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Graphical<\/h3>\r\n<p id=\"fs-id1951802\">For the following exercises, plot the complex numbers on the complex plane.<\/p>\r\n\r\n<div id=\"fs-id2771166\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2771167\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2771166-solution\">11<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2771168\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>1<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>i<\/mi><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>1<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>i<\/mi><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1312026\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1312027\" data-type=\"problem\"><span class=\"os-number\">12<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2048575\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>\u22122<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>\u22122<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1045430\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1403322\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1045430-solution\">13<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1403323\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mi>i<\/mi>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mi>i<\/mi><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2513619\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2513620\" data-type=\"problem\"><span class=\"os-number\">14<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2513621\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>\u22123<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>\u22123<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id2644609\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Numeric<\/h3>\r\n<p id=\"fs-id2777086\">For the following exercises, perform the indicated operation and express the result as a simplified complex number.<\/p>\r\n\r\n<div id=\"fs-id1296020\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1296021\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1296020-solution\">15<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1296022\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>3<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mn>5<\/mn><mo>\u2212<\/mo><mn>3<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>3<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mn>5<\/mn><mo>\u2212<\/mo><mn>3<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1826038\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1826039\" data-type=\"problem\"><span class=\"os-number\">16<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1826040\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22122<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>+<\/mo><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>1<\/mn><mo>+<\/mo><mn>6<\/mn><mi>i<\/mi><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22122<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>+<\/mo><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>1<\/mn><mo>+<\/mo><mn>6<\/mn><mi>i<\/mi><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2441443\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2441444\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2441443-solution\">17<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2778017\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22125<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>6<\/mn><mo>\u2212<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22125<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>6<\/mn><mo>\u2212<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2439091\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2496696\" data-type=\"problem\"><span class=\"os-number\">18<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2496697\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>2<\/mn><mo>\u2212<\/mo><mn>3<\/mn><mi>i<\/mi><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>2<\/mn><mo>\u2212<\/mo><mn>3<\/mn><mi>i<\/mi><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1976046\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1976047\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1976046-solution\">19<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1976048\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>\u22124<\/mn><mo>+<\/mo><mn>4<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u22126<\/mn><mo>+<\/mo><mn>9<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>\u22124<\/mn><mo>+<\/mo><mn>4<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u22126<\/mn><mo>+<\/mo><mn>9<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1509888\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1509889\" data-type=\"problem\"><span class=\"os-number\">20<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1509890\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>2<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>2<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1562525\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1562526\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1562525-solution\">21<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id3176342\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>5<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>i<\/mi><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>5<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>i<\/mi><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2505958\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2267959\" data-type=\"problem\"><span class=\"os-number\">22<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2267960\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>6<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>i<\/mi><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><mn>5<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>6<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>i<\/mi><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><mn>5<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1710164\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1710165\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1710164-solution\">23<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1710166\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22122<\/mn><mo>+<\/mo><mn>4<\/mn><mi>i<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\r\n<mn>8<\/mn>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22122<\/mn><mo>+<\/mo><mn>4<\/mn><mi>i<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\r\n<mn>8<\/mn>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2905259\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1907594\" data-type=\"problem\"><span class=\"os-number\">24<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1907595\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>2<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo>\u2212<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>2<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo>\u2212<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2905821\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2781269\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2905821-solution\">25<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2781270\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22121<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><mn>\u22122<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22121<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><mn>\u22122<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1832281\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1832282\" data-type=\"problem\"><span class=\"os-number\">26<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1832283\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>4<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>i<\/mi><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>4<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>i<\/mi><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1894961\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1894962\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1894961-solution\">27<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1894963\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>3<\/mn><mo>+<\/mo><mn>4<\/mn><mi>i<\/mi><\/mrow>\r\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>3<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>3<\/mn><mo>+<\/mo><mn>4<\/mn><mi>i<\/mi><\/mrow>\r\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>3<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1919888\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1919889\" data-type=\"problem\"><span class=\"os-number\">28<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1919890\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>3<\/mn><mo>+<\/mo><mn>4<\/mn><mi>i<\/mi>\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>3<\/mn><mo>+<\/mo><mn>4<\/mn><mi>i<\/mi>\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1403158\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2293278\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1403158-solution\">29<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2293280\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>6<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>i<\/mi>\r\n<\/mrow>\r\n<mn>3<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>6<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>i<\/mi>\r\n<\/mrow>\r\n<mn>3<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1576927\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1576928\" data-type=\"problem\"><span class=\"os-number\">30<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1726713\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mo>\u2212<\/mo><mn>5<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi>\r\n<\/mrow>\r\n<mrow>\r\n<mn>2<\/mn><mi>i<\/mi>\r\n<\/mrow>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mrow>\r\n<mo>\u2212<\/mo><mn>5<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi>\r\n<\/mrow>\r\n<mrow>\r\n<mn>2<\/mn><mi>i<\/mi>\r\n<\/mrow>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1403309\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1403310\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1403309-solution\">31<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1403311\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>6<\/mn><mo>+<\/mo><mn>4<\/mn><mi>i<\/mi>\r\n<\/mrow>\r\n<mi>i<\/mi>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>6<\/mn><mo>+<\/mo><mn>4<\/mn><mi>i<\/mi>\r\n<\/mrow>\r\n<mi>i<\/mi>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1591150\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1228965\" data-type=\"problem\"><span class=\"os-number\">32<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1228966\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>2<\/mn><mo>\u2212<\/mo><mn>3<\/mn><mi>i<\/mi>\r\n<\/mrow>\r\n<mrow>\r\n<mn>4<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi>\r\n<\/mrow>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>2<\/mn><mo>\u2212<\/mo><mn>3<\/mn><mi>i<\/mi>\r\n<\/mrow>\r\n<mrow>\r\n<mn>4<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi>\r\n<\/mrow>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2437570\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1205233\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2437570-solution\">33<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1205234\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>3<\/mn><mo>+<\/mo><mn>4<\/mn><mi>i<\/mi>\r\n<\/mrow>\r\n<mrow>\r\n<mn>2<\/mn><mo>\u2212<\/mo><mi>i<\/mi>\r\n<\/mrow>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>3<\/mn><mo>+<\/mo><mn>4<\/mn><mi>i<\/mi>\r\n<\/mrow>\r\n<mrow>\r\n<mn>2<\/mn><mo>\u2212<\/mo><mi>i<\/mi>\r\n<\/mrow>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2484840\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2691265\" data-type=\"problem\"><span class=\"os-number\">34<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2691266\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>2<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi>\r\n<\/mrow>\r\n<mrow>\r\n<mn>2<\/mn><mo>\u2212<\/mo><mn>3<\/mn><mi>i<\/mi>\r\n<\/mrow>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>2<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi>\r\n<\/mrow>\r\n<mrow>\r\n<mn>2<\/mn><mo>\u2212<\/mo><mn>3<\/mn><mi>i<\/mi>\r\n<\/mrow>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2506948\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1482316\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2506948-solution\">35<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1482317\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>\u22129<\/mn>\r\n<\/mrow>\r\n<\/msqrt>\r\n<mo>+<\/mo><mn>3<\/mn><msqrt>\r\n<mrow>\r\n<mn>\u221216<\/mn>\r\n<\/mrow>\r\n<\/msqrt>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>\u22129<\/mn>\r\n<\/mrow>\r\n<\/msqrt>\r\n<mo>+<\/mo><mn>3<\/mn><msqrt>\r\n<mrow>\r\n<mn>\u221216<\/mn>\r\n<\/mrow>\r\n<\/msqrt>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2048454\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id3008542\" data-type=\"problem\"><span class=\"os-number\">36<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id3008543\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mo>\u2212<\/mo><msqrt>\r\n<mrow>\r\n<mn>\u22124<\/mn><\/mrow>\r\n<\/msqrt>\r\n<mo>\u2212<\/mo><mn>4<\/mn><msqrt>\r\n<mrow>\r\n<mn>\u221225<\/mn><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mo>\u2212<\/mo><msqrt>\r\n<mrow>\r\n<mn>\u22124<\/mn><\/mrow>\r\n<\/msqrt>\r\n<mo>\u2212<\/mo><mn>4<\/mn><msqrt>\r\n<mrow>\r\n<mn>\u221225<\/mn><\/mrow>\r\n<\/msqrt>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1823209\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1823210\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1823209-solution\">37<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1560121\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>2<\/mn><mo>+<\/mo><msqrt>\r\n<mrow>\r\n<mo>\u2212<\/mo><mn>12<\/mn>\r\n<\/mrow>\r\n<\/msqrt>&nbsp;\r\n\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>2<\/mn><mo>+<\/mo><msqrt>\r\n<mrow>\r\n<mo>\u2212<\/mo><mn>12<\/mn>\r\n<\/mrow>\r\n<\/msqrt>&nbsp;\r\n\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1823150\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2644284\" data-type=\"problem\"><span class=\"os-number\">38<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2644285\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>4<\/mn><mo>+<\/mo><msqrt>\r\n<mrow>\r\n<mo>\u2212<\/mo><mn>20<\/mn>\r\n<\/mrow>\r\n<\/msqrt>&nbsp;\r\n\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>4<\/mn><mo>+<\/mo><msqrt>\r\n<mrow>\r\n<mo>\u2212<\/mo><mn>20<\/mn>\r\n<\/mrow>\r\n<\/msqrt>&nbsp;\r\n\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1690727\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1690728\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1690727-solution\">39<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1690729\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mn>8<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mn>8<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2995714\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2995715\" data-type=\"problem\"><span class=\"os-number\">40<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id3133451\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mrow>\r\n<mn>15<\/mn>\r\n<\/mrow>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mrow>\r\n<mn>15<\/mn>\r\n<\/mrow>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1479782\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2512584\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1479782-solution\">41<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2512585\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mrow>\r\n<mn>22<\/mn>\r\n<\/mrow>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mrow>\r\n<mn>22<\/mn>\r\n<\/mrow>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1537530\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Technology<\/h3>\r\n<p id=\"fs-id1353170\">For the following exercises, use a calculator to help answer the questions.<\/p>\r\n\r\n<div id=\"fs-id2980662\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2980663\" data-type=\"problem\"><span class=\"os-number\">42<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2980664\">Evaluate<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>+<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo>\r\n<\/mrow>\r\n<mi>k<\/mi>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>+<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo>\r\n<\/mrow>\r\n<mi>k<\/mi>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>for\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>k<\/mi><mo>=<\/mo><mn>4<\/mn><mo>,<\/mo><mn>8<\/mn><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>k<\/mi><mo>=<\/mo><mn>4<\/mn><mo>,<\/mo><mn>8<\/mn><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>and\r\n\r\n<math display=\"inline\"><semantics><mrow><mn>12<\/mn><\/mrow><annotation-xml encoding=\"MathML-Content\"><mn>12<\/mn><\/annotation-xml><\/semantics><\/math>. Predict the value if\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>k<\/mi><mo>=<\/mo><mn>16.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>k<\/mi><mo>=<\/mo><mn>16.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2279285\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2279286\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2279285-solution\">43<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2279287\">Evaluate<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo>\r\n<\/mrow>\r\n<mi>k<\/mi>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo>\r\n<\/mrow>\r\n<mi>k<\/mi>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>for\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>k<\/mi><mo>=<\/mo><mn>2<\/mn><mo>,<\/mo><mn>6<\/mn><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>k<\/mi><mo>=<\/mo><mn>2<\/mn><mo>,<\/mo><mn>6<\/mn><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>and\r\n\r\n<math display=\"inline\"><semantics><mrow><mn>10<\/mn><\/mrow><annotation-xml encoding=\"MathML-Content\"><mn>10<\/mn><\/annotation-xml><\/semantics><\/math>. Predict the value if\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>k<\/mi><mo>=<\/mo><mn>14.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>k<\/mi><mo>=<\/mo><mn>14.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1007677\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1007678\" data-type=\"problem\"><span class=\"os-number\">44<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1520536\">Evaluate<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mtext>l<\/mtext><mo>+<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<mi>k<\/mi>\r\n<\/msup>\r\n<mo>\u2212<\/mo><msup>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mtext>l<\/mtext><mo>\u2212<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<mi>k<\/mi>\r\n<\/msup>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mtext>l<\/mtext><mo>+<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<mi>k<\/mi>\r\n<\/msup>\r\n<mo>\u2212<\/mo><msup>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mtext>l<\/mtext><mo>\u2212<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<mi>k<\/mi>\r\n<\/msup>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>for\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>k<\/mi><mo>=<\/mo><mn>4<\/mn><mo>,<\/mo><mn>8<\/mn><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>k<\/mi><mo>=<\/mo><mn>4<\/mn><mo>,<\/mo><mn>8<\/mn><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>and\r\n\r\n<math display=\"inline\"><semantics><mrow><mn>12<\/mn><\/mrow><annotation-xml encoding=\"MathML-Content\"><mn>12<\/mn><\/annotation-xml><\/semantics><\/math>. Predict the value for\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>k<\/mi><mo>=<\/mo><mn>16.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>k<\/mi><mo>=<\/mo><mn>16.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2410330\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2410331\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2410330-solution\">45<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2410332\">Show that a solution of<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>6<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>1<\/mn><mo>=<\/mo><mn>0<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>6<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>1<\/mn><mo>=<\/mo><mn>0<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>is\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<msqrt>\r\n<mn>3<\/mn>\r\n<\/msqrt>&nbsp;\r\n\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mo>+<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mi>i<\/mi><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mrow>\r\n<msqrt>\r\n<mn>3<\/mn>\r\n<\/msqrt>&nbsp;\r\n\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mo>+<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mi>i<\/mi><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1385742\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1385743\" data-type=\"problem\"><span class=\"os-number\">46<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1385744\">Show that a solution of<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>8<\/mn>\r\n<\/msup>\r\n<mn>\u22121<\/mn><mo>=<\/mo><mn>0<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>8<\/mn>\r\n<\/msup>\r\n<mn>\u22121<\/mn><mo>=<\/mo><mn>0<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>is\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<msqrt>\r\n<mn>2<\/mn>\r\n<\/msqrt>&nbsp;\r\n\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mo>+<\/mo><mfrac>\r\n<mrow>\r\n<msqrt>\r\n<mn>2<\/mn>\r\n<\/msqrt>&nbsp;\r\n\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mi>i<\/mi><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mrow>\r\n<msqrt>\r\n<mn>2<\/mn>\r\n<\/msqrt>&nbsp;\r\n\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mo>+<\/mo><mfrac>\r\n<mrow>\r\n<msqrt>\r\n<mn>2<\/mn>\r\n<\/msqrt>&nbsp;\r\n\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mi>i<\/mi><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id2494010\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Extensions<\/h3>\r\n<p id=\"fs-id2371706\">For the following exercises, evaluate the expressions, writing the result as a simplified complex number.<\/p>\r\n\r\n<div id=\"fs-id3161030\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id3161031\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id3161030-solution\">47<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2434972\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mi>i<\/mi>\r\n<\/mfrac>\r\n<mo>+<\/mo><mfrac>\r\n<mn>4<\/mn>\r\n<mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mn>3<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mi>i<\/mi>\r\n<\/mfrac>\r\n<mo>+<\/mo><mfrac>\r\n<mn>4<\/mn>\r\n<mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mn>3<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2794969\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2794970\" data-type=\"problem\"><span class=\"os-number\">48<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2794971\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mrow>\r\n<mn>11<\/mn>\r\n<\/mrow>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mrow>\r\n<mn>21<\/mn>\r\n<\/mrow>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mrow>\r\n<mn>11<\/mn>\r\n<\/mrow>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mrow>\r\n<mn>21<\/mn>\r\n<\/mrow>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2716062\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2716064\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2716062-solution\">49<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2515854\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mn>7<\/mn>\r\n<\/msup>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>1<\/mn><mo>+<\/mo><msup>\r\n<mi>i<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mn>7<\/mn>\r\n<\/msup>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>1<\/mn><mo>+<\/mo><msup>\r\n<mi>i<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2370481\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2943044\" data-type=\"problem\"><span class=\"os-number\">50<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2943045\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mrow>\r\n<mn>\u22123<\/mn>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>+<\/mo><mn>5<\/mn><msup>\r\n<mi>i<\/mi>\r\n<mn>7<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>i<\/mi>\r\n<mrow>\r\n<mn>\u22123<\/mn>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>+<\/mo><mn>5<\/mn><msup>\r\n<mi>i<\/mi>\r\n<mn>7<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1390725\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1390726\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1390725-solution\">51<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1390727\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>2<\/mn><mo>+<\/mo><mi>i<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>4<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>i<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>+<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo>\r\n<\/mrow>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>2<\/mn><mo>+<\/mo><mi>i<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>4<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>i<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>+<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo>\r\n<\/mrow>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1914286\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1914287\" data-type=\"problem\"><span class=\"os-number\">52<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1945289\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>1<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi><\/mrow>\r\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>2<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>1<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi><\/mrow>\r\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>2<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mfrac>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2357467\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2357468\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2357467-solution\">53<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2357469\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>3<\/mn><mo>+<\/mo><mi>i<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<mrow>\r\n<msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>1<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mrow>\r\n<msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>3<\/mn><mo>+<\/mo><mi>i<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<mrow>\r\n<msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>1<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2294729\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2721846\" data-type=\"problem\"><span class=\"os-number\">54<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2721848\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>3<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi>\r\n<\/mrow>\r\n<mrow>\r\n<mn>2<\/mn><mo>+<\/mo><mi>i<\/mi>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>+<\/mo><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>4<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>3<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi>\r\n<\/mrow>\r\n<mrow>\r\n<mn>2<\/mn><mo>+<\/mo><mi>i<\/mi>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>+<\/mo><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>4<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1758544\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1758545\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1758544-solution\">55<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2932432\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>4<\/mn><mo>+<\/mo><mi>i<\/mi>\r\n<\/mrow>\r\n<mi>i<\/mi>\r\n<\/mfrac>\r\n<mo>+<\/mo><mfrac>\r\n<mrow>\r\n<mn>3<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi>\r\n<\/mrow>\r\n<mrow>\r\n<mn>1<\/mn><mo>\u2212<\/mo><mi>i<\/mi>\r\n<\/mrow>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>4<\/mn><mo>+<\/mo><mi>i<\/mi>\r\n<\/mrow>\r\n<mi>i<\/mi>\r\n<\/mfrac>\r\n<mo>+<\/mo><mfrac>\r\n<mrow>\r\n<mn>3<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi>\r\n<\/mrow>\r\n<mrow>\r\n<mn>1<\/mn><mo>\u2212<\/mo><mi>i<\/mi>\r\n<\/mrow>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id3034376\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id3034377\" data-type=\"problem\"><span class=\"os-number\">56<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2402604\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>3<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi>\r\n<\/mrow>\r\n<mrow>\r\n<mn>1<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mrow>\r\n<mn>2<\/mn><mo>\u2212<\/mo><mn>3<\/mn><mi>i<\/mi>\r\n<\/mrow>\r\n<mrow>\r\n<mn>3<\/mn><mo>+<\/mo><mi>i<\/mi>\r\n<\/mrow>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>3<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi>\r\n<\/mrow>\r\n<mrow>\r\n<mn>1<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mrow>\r\n<mn>2<\/mn><mo>\u2212<\/mo><mn>3<\/mn><mi>i<\/mi>\r\n<\/mrow>\r\n<mrow>\r\n<mn>3<\/mn><mo>+<\/mo><mi>i<\/mi>\r\n<\/mrow>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/section><\/div>\r\n<\/div>\r\n<\/div>","rendered":"<div id=\"main-content\" class=\"MainContent__ContentStyles-sc-6yy1if-0 NnXKu\" tabindex=\"-1\" data-dynamic-style=\"true\">\n<div id=\"page_4aa9351c-019f-4c06-bb40-d58262ea7ec7\" class=\"chapter-content-module\" data-type=\"page\" data-book-content=\"true\">\n<div class=\"ui-has-child-title\" data-type=\"abstract\">\n<header>\n<h2 data-type=\"title\">Learning Objectives<\/h2>\n<\/header>\n<section>\n<p id=\"para-00001\">In this section, you will:<\/p>\n<ul id=\"list-00001\">\n<li>Add and subtract complex numbers.<\/li>\n<li>Multiply and divide complex numbers.<\/li>\n<li>Simplify powers of<br \/>\n<math display=\"inline\"><semantics><mrow><mi>i<\/mi><\/mrow><annotation-xml encoding=\"MathML-Content\"><mi>i<\/mi><\/annotation-xml><\/semantics><\/math><\/li>\n<\/ul>\n<\/section>\n<\/div>\n<div id=\"Figure_02_04_001\" class=\"os-figure\">\n<figure class=\"medium\" data-id=\"Figure_02_04_001\"><span id=\"fs-id1919899\" data-type=\"media\" data-alt=\"A visual representation of the Mandelbrot set\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/openstax.org\/books\/college-algebra-2e\/pages\/src#fixme\" alt=\"A visual representation of the Mandelbrot set\" width=\"437\" height=\"325\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/250825d36367ccdee3e00cbf20958515aa106674\" \/><br \/>\n<\/span><\/figure>\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Figure <\/span><br \/>\n<span class=\"os-number\">1<\/span><\/div>\n<\/div>\n<p id=\"fs-id2523388\">Discovered by Benoit Mandelbrot around 1980, the Mandelbrot Set is one of the most recognizable fractal images. The image is built on the theory of self-similarity and the operation of iteration. Zooming in on a fractal image brings many surprises, particularly in the high level of repetition of detail that appears as magnification increases. The equation that generates this image turns out to be rather simple.<\/p>\n<p id=\"fs-id2510203\">In order to better understand it, we need to become familiar with a new set of numbers. Keep in mind that the study of mathematics continuously builds upon itself. Negative integers, for example, fill a void left by the set of positive integers. The set of rational numbers, in turn, fills a void left by the set of integers. The set of real numbers fills a void left by the set of rational numbers. Not surprisingly, the set of real numbers has voids as well. In this section, we will explore a set of numbers that fills voids in the set of real numbers and find out how to work within it.<\/p>\n<section id=\"fs-id2425607\" data-depth=\"1\">\n<h2 data-type=\"title\">Expressing Square Roots of Negative Numbers as Multiples of <math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>i<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>i<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/h2>\n<p id=\"fs-id1385456\">We know how to find the square root of any positive real number. In a similar way, we can find the square root of any negative number. The difference is that the root is not real. If the value in the radicand is negative, the root is said to be an imaginary number<strong>. <\/strong>The imaginary number<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>i<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>i<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>is defined as the square root of<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>\u22121.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>\u22121.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<div id=\"fs-id1276418\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<msqrt>\n<mrow>\n<mn>\u22121<\/mn>\n<\/mrow>\n<\/msqrt>\n<mo>=<\/mo><mi>i<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msqrt>\n<mrow>\n<mn>\u22121<\/mn>\n<\/mrow>\n<\/msqrt>\n<mo>=<\/mo><mi>i<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id3155144\">So, using properties of radicals,<\/p>\n<div id=\"fs-id2005251\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>i<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>=<\/mo><msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<msqrt>\n<mrow>\n<mo>\u2212<\/mo><mn>1<\/mn>\n<\/mrow>\n<\/msqrt>&nbsp;\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<mo>=<\/mo><mn>\u22121<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>i<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>=<\/mo><msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<msqrt>\n<mrow>\n<mo>\u2212<\/mo><mn>1<\/mn>\n<\/mrow>\n<\/msqrt>&nbsp;\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<mo>=<\/mo><mn>\u22121<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1523248\">We can write the square root of any negative number as a multiple of<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>i<\/mi><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>i<\/mi><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>Consider the square root of<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>\u221249.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>\u221249.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<div id=\"fs-id1296283\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><msqrt><mrow><mn>\u221249<\/mn><\/mrow><\/msqrt><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<msqrt>\n<mrow>\n<mn>49<\/mn><mo>\u22c5<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u22121<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<msqrt>\n<mrow>\n<mn>49<\/mn><\/mrow>\n<\/msqrt>\n<msqrt>\n<mrow>\n<mn>\u22121<\/mn><\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>7<\/mn><mi>i<\/mi><\/mrow><\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><msqrt><mrow><mn>\u221249<\/mn><\/mrow><\/msqrt><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<msqrt>\n<mrow>\n<mn>49<\/mn><mo>\u22c5<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u22121<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<msqrt>\n<mrow>\n<mn>49<\/mn><\/mrow>\n<\/msqrt>\n<msqrt>\n<mrow>\n<mn>\u22121<\/mn><\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>7<\/mn><mi>i<\/mi><\/mrow><\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1149119\">We use<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>7<\/mn><mi>i<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>7<\/mn><mi>i<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>and not<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>\u22127<\/mn><mi>i<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>\u22127<\/mn><mi>i<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>because the principal root of<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>49<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>49<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>is the positive root.<\/p>\n<p id=\"fs-id2708516\">A complex number is the sum of a real number and an imaginary number. A complex number is expressed in standard form when written<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>where<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>a<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>a<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>is the real part and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>b<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>b<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>is the imaginary part. For example,<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>5<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>5<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>is a complex number. So, too, is<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>3<\/mn><mo>+<\/mo><mn>4<\/mn><mi>i<\/mi><msqrt>\n<mn>3<\/mn>\n<\/msqrt>\n<mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>3<\/mn><mo>+<\/mo><mn>4<\/mn><mi>i<\/mi><msqrt>\n<mn>3<\/mn>\n<\/msqrt>\n<mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<p><span id=\"fs-id1541885\" data-type=\"media\" data-alt=\"The complex number 5 + 2i is displayed. The 5 is labeled as: Real part and the 2i is labeled as: Imaginary part\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/openstax.org\/books\/college-algebra-2e\/pages\/src#fixme\" alt=\"The complex number 5 + 2i is displayed. The 5 is labeled as: Real part and the 2i is labeled as: Imaginary part\" width=\"487\" height=\"76\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/b6771cfacdac806d7546b9b1d376853f0f822803\" \/><br \/>\n<\/span><\/p>\n<p id=\"fs-id1174676\">Imaginary numbers differ from real numbers in that a squared imaginary number produces a negative real number. Recall that when a positive real number is squared, the result is a positive real number and when a negative real number is squared, the result is also a positive real number. Complex numbers consist of real and imaginary numbers.<\/p>\n<div id=\"fs-id1558068\" class=\"ui-has-child-title\" data-type=\"note\">\n<header>\n<h2 class=\"os-title\" data-type=\"title\"><span class=\"os-title-label\" data-type=\"\">Imaginary and Complex Numbers<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<p id=\"fs-id1689292\">A <span id=\"term-00002\" data-type=\"term\">complex number<\/span> is a number of the form<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>where<\/p>\n<ul id=\"fs-id1531123\">\n<li><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>a<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>a<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>is the real part of the complex number.<\/li>\n<li><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>b<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>b<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>is the imaginary part of the complex number.<\/li>\n<\/ul>\n<p id=\"fs-id1402821\">If<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>b<\/mi><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>b<\/mi><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>then<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>is a real number. If<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>a<\/mi><mo>=<\/mo><mn>0<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>a<\/mi><mo>=<\/mo><mn>0<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>b<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>b<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>is not equal to 0, the complex number is called a pure imaginary number. An <span id=\"term-00003\" data-type=\"term\">imaginary number<\/span> is an even root of a negative number.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1266532\" class=\"how-to-notitle ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"How To\">\n<header>\n<h2 class=\"os-title\" data-type=\"title\" data-label-parent=\"How To\"><span class=\"os-title-label\">How To<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<p id=\"fs-id1578095\"><strong>Given an imaginary number, express it in the standard form of a complex number.<\/strong><\/p>\n<ol id=\"fs-id1269513\" type=\"1\">\n<li>Write<br \/>\n<math display=\"inline\"><semantics><mrow>\n<mrow>\n<msqrt>\n<mrow>\n<mo>\u2212<\/mo><mi>a<\/mi>\n<\/mrow>\n<\/msqrt>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msqrt>\n<mrow>\n<mo>\u2212<\/mo><mi>a<\/mi>\n<\/mrow>\n<\/msqrt>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math>as<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msqrt>\n<mi>a<\/mi>\n<\/msqrt>\n<msqrt>\n<mrow>\n<mn>\u22121<\/mn>\n<\/mrow>\n<\/msqrt>\n<mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msqrt>\n<mi>a<\/mi>\n<\/msqrt>\n<msqrt>\n<mrow>\n<mn>\u22121<\/mn>\n<\/mrow>\n<\/msqrt>\n<mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\n<li>Express<br \/>\n<math display=\"inline\"><semantics><mrow>\n<mrow>\n<msqrt>\n<mrow>\n<mn>\u22121<\/mn>\n<\/mrow>\n<\/msqrt>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msqrt>\n<mrow>\n<mn>\u22121<\/mn>\n<\/mrow>\n<\/msqrt>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math>as<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>i<\/mi><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>i<\/mi><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\n<li>Write<br \/>\n<math display=\"inline\"><semantics><mrow>\n<mrow>\n<msqrt>\n<mi>a<\/mi>\n<\/msqrt>\n<mo>\u22c5<\/mo><mi>i<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msqrt>\n<mi>a<\/mi>\n<\/msqrt>\n<mo>\u22c5<\/mo><mi>i<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>in simplest form.<\/li>\n<\/ol>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"Example_02_04_01\" class=\"ui-has-child-title\" data-type=\"example\">\n<header>\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span><br \/>\n<span class=\"os-number\">1<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"body\">\n<div id=\"fs-id1884617\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1228287\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<h3 data-type=\"title\">Expressing an Imaginary Number in Standard Form<\/h3>\n<p id=\"fs-id1274827\">Express<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msqrt>\n<mrow>\n<mn>\u22129<\/mn>\n<\/mrow>\n<\/msqrt>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msqrt>\n<mrow>\n<mn>\u22129<\/mn>\n<\/mrow>\n<\/msqrt>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math>in standard form.<\/p>\n<\/div>\n<\/div>\n<details id=\"fs-id1813876\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\">\n<summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary>\n<section class=\"ui-body\" role=\"alert\">\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\n<div class=\"os-solution-container\">\n<div id=\"fs-id2443487\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><msqrt><mrow><mn>\u22129<\/mn><\/mrow><\/msqrt><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<msqrt>\n<mn>9<\/mn>\n<\/msqrt>\n<msqrt>\n<mrow>\n<mn>\u22121<\/mn><\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mn>3<\/mn><mi>i<\/mi><\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><msqrt><mrow><mn>\u22129<\/mn><\/mrow><\/msqrt><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<msqrt>\n<mn>9<\/mn>\n<\/msqrt>\n<msqrt>\n<mrow>\n<mn>\u22121<\/mn><\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mn>3<\/mn><mi>i<\/mi><\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id881430\">In standard form, this is<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>0<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>0<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/details>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2375806\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\">\n<header>\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span><br \/>\n<span class=\"os-number\">#1<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<div id=\"ti_02_04_01\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1294134\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"fs-id3048344\">Express<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msqrt>\n<mrow>\n<mn>\u221224<\/mn>\n<\/mrow>\n<\/msqrt>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msqrt>\n<mrow>\n<mn>\u221224<\/mn>\n<\/mrow>\n<\/msqrt>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math>in standard form.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1591834\" data-depth=\"1\">\n<h2 data-type=\"title\">Plotting a Complex Number on the Complex Plane<\/h2>\n<p id=\"fs-id3176103\">We cannot plot complex numbers on a number line as we might real numbers. However, we can still represent them graphically. To represent a complex number, we need to address the two components of the number. We use the <span id=\"term-00004\" data-type=\"term\">complex plane<\/span>, which is a coordinate system in which the horizontal axis represents the real component and the vertical axis represents the imaginary component. Complex numbers are the points on the plane, expressed as ordered pairs<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>,<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>,<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>where<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>a<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>a<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>represents the coordinate for the horizontal axis and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>b<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>b<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>represents the coordinate for the vertical axis.<\/p>\n<p id=\"fs-id2517292\">Let\u2019s consider the number<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>\u22122<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>\u22122<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>The real part of the complex number is<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>\u22122<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>\u22122<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>and the imaginary part is 3. We plot the ordered pair<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>\u22122<\/mn><mo>,<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mo stretchy=\"false\">(<\/mo><mn>\u22122<\/mn><mo>,<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>to represent the complex number<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>\u22122<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>\u22122<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>as shown in <a class=\"autogenerated-content\" href=\"2-4-complex-numbers#Figure_02_04_003\">Figure 2<\/a><strong>.<\/strong><\/p>\n<div id=\"Figure_02_04_003\" class=\"os-figure\">\n<figure class=\"small\" data-id=\"Figure_02_04_003\"><span id=\"fs-id1245266\" data-type=\"media\" data-alt=\"Coordinate plane with the x and y axes ranging from negative 5 to 5. The point negative 2 plus 3i is plotted on the graph. An arrow extends leftward from the origin two units and then an arrow extends upward three units from the end of the previous arrow.\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/openstax.org\/books\/college-algebra-2e\/pages\/src#fixme\" alt=\"Coordinate plane with the x and y axes ranging from negative 5 to 5. The point negative 2 plus 3i is plotted on the graph. An arrow extends leftward from the origin two units and then an arrow extends upward three units from the end of the previous arrow.\" width=\"357\" height=\"372\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/4bc24ed9c0f41a9c1eb3faf6eea40b42241d503b\" \/><br \/>\n<\/span><\/figure>\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Figure <\/span><br \/>\n<span class=\"os-number\">2<\/span><\/div>\n<\/div>\n<div id=\"fs-id1184889\" class=\"ui-has-child-title\" data-type=\"note\">\n<header>\n<h2 class=\"os-title\" data-type=\"title\"><span class=\"os-title-label\" data-type=\"\">Complex Plane<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<p id=\"fs-id3269164\">In the complex plane, the horizontal axis is the real axis, and the vertical axis is the imaginary axis, as shown in <a class=\"autogenerated-content\" href=\"2-4-complex-numbers#Figure_02_04_004\">Figure 3<\/a>.<\/p>\n<div id=\"Figure_02_04_004\" class=\"os-figure\">\n<figure class=\"small\" data-id=\"Figure_02_04_004\"><span id=\"fs-id1943089\" data-type=\"media\" data-alt=\"A blank coordinate plane with the x-axis labeled: real and the y-axis labeled: imaginary.\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/openstax.org\/books\/college-algebra-2e\/pages\/src#fixme\" alt=\"A blank coordinate plane with the x-axis labeled: real and the y-axis labeled: imaginary.\" width=\"229\" height=\"208\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/f597ea5f207951735a0de498e02a07e8cd28e3f5\" \/><br \/>\n<\/span><\/figure>\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Figure <\/span><br \/>\n<span class=\"os-number\">3<\/span><\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1273812\" class=\"how-to-notitle ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"How To\">\n<header>\n<h2 class=\"os-title\" data-type=\"title\" data-label-parent=\"How To\"><span class=\"os-title-label\">How To<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<p id=\"fs-id1887259\"><strong>Given a complex number, represent its components on the complex plane.<\/strong><\/p>\n<ol id=\"fs-id2484938\" type=\"1\">\n<li>Determine the real part and the imaginary part of the complex number.<\/li>\n<li>Move along the horizontal axis to show the real part of the number.<\/li>\n<li>Move parallel to the vertical axis to show the imaginary part of the number.<\/li>\n<li>Plot the point.<\/li>\n<\/ol>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"Example_02_04_02\" class=\"ui-has-child-title\" data-type=\"example\">\n<header>\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span><br \/>\n<span class=\"os-number\">2<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"body\">\n<div id=\"fs-id3113300\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1787353\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<h3 data-type=\"title\">Plotting a Complex Number on the Complex Plane<\/h3>\n<p id=\"fs-id1516111\">Plot the complex number<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>3<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi><mspace width=\"0.5em\"><\/mspace><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>3<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi><mspace width=\"0.5em\"><\/mspace><\/mrow><\/annotation-xml><\/semantics><\/math>on the complex plane.<\/p>\n<\/div>\n<\/div>\n<details id=\"fs-id1148988\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\">\n<summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary>\n<section class=\"ui-body\" role=\"alert\">\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\n<div class=\"os-solution-container\">\n<p id=\"fs-id2438175\">The real part of the complex number is<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>3<\/mn><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>3<\/mn><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>and the imaginary part is \u20134. We plot the ordered pair<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mo>,<\/mo><mn>\u22124<\/mn><mo stretchy=\"false\">)<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mo>,<\/mo><mn>\u22124<\/mn><mo stretchy=\"false\">)<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>as shown in <a class=\"autogenerated-content\" href=\"2-4-complex-numbers#Figure_02_04_005\">Figure 4<\/a><strong>.<\/strong><\/p>\n<div id=\"Figure_02_04_005\" class=\"os-figure\">\n<figure class=\"small\" data-id=\"Figure_02_04_005\"><span id=\"fs-id915324\" data-type=\"media\" data-alt=\"Coordinate plane with the x and y axes ranging from -5 to 5. The point 3 \u2013 4i is plotted, with an arrow extending rightward from the origin 3 units and an arrow extending downward 4 units from the end of the previous arrow.\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/openstax.org\/books\/college-algebra-2e\/pages\/src#fixme\" alt=\"Coordinate plane with the x and y axes ranging from -5 to 5. The point 3 \u2013 4i is plotted, with an arrow extending rightward from the origin 3 units and an arrow extending downward 4 units from the end of the previous arrow.\" width=\"357\" height=\"372\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/5cad9c29f707a8f3cd1716c1288a2ca1eb8963aa\" \/><br \/>\n<\/span><\/figure>\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Figure <\/span><br \/>\n<span class=\"os-number\">4<\/span><\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/details>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2512646\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\">\n<header>\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span><br \/>\n<span class=\"os-number\">#2<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<div id=\"ti_02_04_02\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1593307\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"fs-id2507797\">Plot the complex number<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>\u22124<\/mn><mo>\u2212<\/mo><mi>i<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>\u22124<\/mn><mo>\u2212<\/mo><mi>i<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>on the complex plane.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1424098\" data-depth=\"1\">\n<h2 data-type=\"title\">Adding and Subtracting Complex Numbers<\/h2>\n<p id=\"fs-id1463014\">Just as with real numbers, we can perform arithmetic operations on complex numbers. To add or subtract complex numbers, we combine the real parts and then combine the imaginary parts.<\/p>\n<div id=\"fs-id1442785\" class=\"ui-has-child-title\" data-type=\"note\">\n<header>\n<h2 class=\"os-title\" data-type=\"title\"><span class=\"os-title-label\" data-type=\"\">Complex Numbers: Addition and Subtraction<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<p id=\"fs-id1976732\">Adding complex numbers:<\/p>\n<div id=\"fs-id1519513\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>+<\/mo><mrow><mo>(<\/mo>\n<mrow>\n<mi>c<\/mi><mo>+<\/mo><mi>d<\/mi><mi>i<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>=<\/mo><mrow><mo>(<\/mo>\n<mrow>\n<mi>a<\/mi><mo>+<\/mo><mi>c<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>+<\/mo><mrow><mo>(<\/mo>\n<mrow>\n<mi>b<\/mi><mo>+<\/mo><mi>d<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mi>i<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>+<\/mo><mrow><mo>(<\/mo>\n<mrow>\n<mi>c<\/mi><mo>+<\/mo><mi>d<\/mi><mi>i<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>=<\/mo><mrow><mo>(<\/mo>\n<mrow>\n<mi>a<\/mi><mo>+<\/mo><mi>c<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>+<\/mo><mrow><mo>(<\/mo>\n<mrow>\n<mi>b<\/mi><mo>+<\/mo><mi>d<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mi>i<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1446903\">Subtracting complex numbers:<\/p>\n<div id=\"fs-id1018786\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>\u2212<\/mo><mrow><mo>(<\/mo>\n<mrow>\n<mi>c<\/mi><mo>+<\/mo><mi>d<\/mi><mi>i<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>=<\/mo><mrow><mo>(<\/mo>\n<mrow>\n<mi>a<\/mi><mo>\u2212<\/mo><mi>c<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>+<\/mo><mrow><mo>(<\/mo>\n<mrow>\n<mi>b<\/mi><mo>\u2212<\/mo><mi>d<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mi>i<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>\u2212<\/mo><mrow><mo>(<\/mo>\n<mrow>\n<mi>c<\/mi><mo>+<\/mo><mi>d<\/mi><mi>i<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>=<\/mo><mrow><mo>(<\/mo>\n<mrow>\n<mi>a<\/mi><mo>\u2212<\/mo><mi>c<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>+<\/mo><mrow><mo>(<\/mo>\n<mrow>\n<mi>b<\/mi><mo>\u2212<\/mo><mi>d<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mi>i<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1296613\" class=\"how-to-notitle ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"How To\">\n<header>\n<h2 class=\"os-title\" data-type=\"title\" data-label-parent=\"How To\"><span class=\"os-title-label\">How To<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<p id=\"fs-id2383706\"><strong>Given two complex numbers, find the sum or difference.<\/strong><\/p>\n<ol id=\"fs-id1805520\" type=\"1\">\n<li>Identify the real and imaginary parts of each number.<\/li>\n<li>Add or subtract the real parts.<\/li>\n<li>Add or subtract the imaginary parts.<\/li>\n<\/ol>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"Example_02_04_03\" class=\"ui-has-child-title\" data-type=\"example\">\n<header>\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span><br \/>\n<span class=\"os-number\">3<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"body\">\n<div id=\"fs-id1133706\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1267465\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<h3 data-type=\"title\">Adding and Subtracting Complex Numbers<\/h3>\n<p id=\"fs-id1567983\">Add or subtract as indicated.<\/p>\n<ol id=\"fs-id3257388\" class=\"circled\" type=\"1\">\n<li><span class=\"token\">\u24d0<\/span><br \/>\n<math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>3<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi><\/mrow>\n<mo>)<\/mo><\/mrow><mo>+<\/mo><mrow><mo>(<\/mo>\n<mrow>\n<mn>2<\/mn><mo>+<\/mo><mn>5<\/mn><mi>i<\/mi><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>3<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi><\/mrow>\n<mo>)<\/mo><\/mrow><mo>+<\/mo><mrow><mo>(<\/mo>\n<mrow>\n<mn>2<\/mn><mo>+<\/mo><mn>5<\/mn><mi>i<\/mi><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\n<li><span class=\"token\">\u24d1<\/span><br \/>\n<math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>\u22125<\/mn><mo>+<\/mo><mn>7<\/mn><mi>i<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>\u2212<\/mo><mrow><mo>(<\/mo>\n<mrow>\n<mn>\u221211<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>\u22125<\/mn><mo>+<\/mo><mn>7<\/mn><mi>i<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>\u2212<\/mo><mrow><mo>(<\/mo>\n<mrow>\n<mn>\u221211<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\n<\/ol>\n<\/div>\n<\/div>\n<details id=\"fs-id1429773\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\">\n<summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary>\n<section class=\"ui-body\" role=\"alert\">\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\n<div class=\"os-solution-container\">\n<p id=\"fs-id1288146\">We add the real parts and add the imaginary parts.<\/p>\n<ol id=\"fs-id2710031\" class=\"circled\" type=\"1\">\n<li><span class=\"token\">\u24d0<\/span><span data-type=\"newline\"><br \/>\n<\/span><br \/>\n<math display=\"inline\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo>+<\/mo><mn>5<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow>\n<mn>3<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi><mo>+<\/mo><mn>2<\/mn><mo>+<\/mo><mn>5<\/mn><mi>i<\/mi><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mn>3<\/mn><mo>+<\/mo><mn>2<\/mn><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u22124<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>5<\/mn><mi>i<\/mi><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mo>+<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u22124<\/mn><mo>+<\/mo><mn>5<\/mn><mo stretchy=\"false\">)<\/mo><mi>i<\/mi><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>5<\/mn><mo>+<\/mo><mi>i<\/mi><\/mrow><\/mtd>\n<\/mtr>\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo>+<\/mo><mn>5<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow>\n<mn>3<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi><mo>+<\/mo><mn>2<\/mn><mo>+<\/mo><mn>5<\/mn><mi>i<\/mi><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mn>3<\/mn><mo>+<\/mo><mn>2<\/mn><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u22124<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>5<\/mn><mi>i<\/mi><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mo>+<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u22124<\/mn><mo>+<\/mo><mn>5<\/mn><mo stretchy=\"false\">)<\/mo><mi>i<\/mi><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>5<\/mn><mo>+<\/mo><mi>i<\/mi><\/mrow><\/mtd>\n<\/mtr>\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\n<li><span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/>\n<\/span><br \/>\n<math display=\"inline\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>\u22125<\/mn><mo>+<\/mo><mn>7<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u221211<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>\u22125<\/mn><mo>+<\/mo><mn>7<\/mn><mi>i<\/mi><mo>+<\/mo><mn>11<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>i<\/mi><\/mrow><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>\u22125<\/mn><mo>+<\/mo><mn>11<\/mn><mo>+<\/mo><mn>7<\/mn><mi>i<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mi>i<\/mi><\/mrow><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mo stretchy=\"false\">(<\/mo><mn>\u22125<\/mn><mo>+<\/mo><mn>11<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mn>7<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><mi>i<\/mi><\/mrow><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>6<\/mn><mo>+<\/mo><mn>5<\/mn><mi>i<\/mi><\/mrow><\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>\u22125<\/mn><mo>+<\/mo><mn>7<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u221211<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>\u22125<\/mn><mo>+<\/mo><mn>7<\/mn><mi>i<\/mi><mo>+<\/mo><mn>11<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>i<\/mi><\/mrow><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>\u22125<\/mn><mo>+<\/mo><mn>11<\/mn><mo>+<\/mo><mn>7<\/mn><mi>i<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mi>i<\/mi><\/mrow><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mo stretchy=\"false\">(<\/mo><mn>\u22125<\/mn><mo>+<\/mo><mn>11<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mn>7<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><mi>i<\/mi><\/mrow><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>6<\/mn><mo>+<\/mo><mn>5<\/mn><mi>i<\/mi><\/mrow><\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\n<\/ol>\n<\/div>\n<\/section>\n<\/details>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2302121\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\">\n<header>\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span><br \/>\n<span class=\"os-number\">#3<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<div id=\"ti_02_04_03\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1339021\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"fs-id1334125\">Subtract<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>2<\/mn><mo>+<\/mo><mn>5<\/mn><mi>i<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>2<\/mn><mo>+<\/mo><mn>5<\/mn><mi>i<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>from<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>3<\/mn><mo>\u2013<\/mo><mn>4<\/mn><mi>i<\/mi><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>3<\/mn><mo>\u2013<\/mo><mn>4<\/mn><mi>i<\/mi><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1322676\" data-depth=\"1\">\n<h2 data-type=\"title\">Multiplying Complex Numbers<\/h2>\n<p id=\"fs-id1188861\">Multiplying complex numbers is much like multiplying binomials. The major difference is that we work with the real and imaginary parts separately.<\/p>\n<section id=\"fs-id2995792\" data-depth=\"2\">\n<h3 data-type=\"title\">Multiplying a Complex Number by a Real Number<\/h3>\n<p id=\"fs-id1533933\">Lets begin by multiplying a complex number by a real number. We distribute the real number just as we would with a binomial. Consider, for example,<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>3<\/mn><mrow><mo>(<\/mo>\n<mrow>\n<mn>6<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>3<\/mn><mrow><mo>(<\/mo>\n<mrow>\n<mn>6<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow><\/annotation-xml><\/semantics><\/math>:<\/p>\n<p><span id=\"fs-id1545734\" data-type=\"media\" data-alt=\"Multiplication of a real number and a complex number. The 3 outside of the parentheses has arrows extending from it to both the 6 and the 2i inside of the parentheses. This expression is set equal to the quantity three times six plus the quantity three times two times i; this is the distributive property. The next line equals eighteen plus six times i; the simplification.\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/openstax.org\/books\/college-algebra-2e\/pages\/src#fixme\" alt=\"Multiplication of a real number and a complex number. The 3 outside of the parentheses has arrows extending from it to both the 6 and the 2i inside of the parentheses. This expression is set equal to the quantity three times six plus the quantity three times two times i; this is the distributive property. The next line equals eighteen plus six times i; the simplification.\" width=\"487\" height=\"87\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/cc0c0bc49d1474d8c158cb4600642d565a51fa2b\" \/><br \/>\n<\/span><\/p>\n<div id=\"fs-id1494245\" class=\"how-to-notitle ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"How To\">\n<header>\n<h2 class=\"os-title\" data-type=\"title\" data-label-parent=\"How To\"><span class=\"os-title-label\">How To<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<p id=\"fs-id2714950\"><strong>Given a complex number and a real number, multiply to find the product.<\/strong><\/p>\n<ol id=\"fs-id1170540\" type=\"1\">\n<li>Use the distributive property.<\/li>\n<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"Example_02_04_04\" class=\"ui-has-child-title\" data-type=\"example\">\n<header>\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span><br \/>\n<span class=\"os-number\">4<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"body\">\n<div id=\"fs-id1203991\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1930439\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<h3 data-type=\"title\">Multiplying a Complex Number by a Real Number<\/h3>\n<p id=\"fs-id1846156\">Find the product<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>4<\/mn><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo>+<\/mo><mn>5<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>4<\/mn><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo>+<\/mo><mn>5<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<details id=\"fs-id2664861\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\">\n<summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary>\n<section class=\"ui-body\" role=\"alert\">\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\n<div class=\"os-solution-container\">\n<p id=\"fs-id1509986\">Distribute the 4.<\/p>\n<div id=\"fs-id1133872\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mn>4<\/mn><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo>+<\/mo><mn>5<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo>\u22c5<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo>\u22c5<\/mo><mn>5<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>8<\/mn><mo>+<\/mo><mn>20<\/mn><mi>i<\/mi><\/mrow><\/mtd>\n<\/mtr>\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mn>4<\/mn><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo>+<\/mo><mn>5<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo>\u22c5<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo>\u22c5<\/mo><mn>5<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>8<\/mn><mo>+<\/mo><mn>20<\/mn><mi>i<\/mi><\/mrow><\/mtd>\n<\/mtr>\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<\/div>\n<\/section>\n<\/details>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1745333\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\">\n<header>\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span><br \/>\n<span class=\"os-number\">#4<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<div id=\"ti_02_04_04\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1211448\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"fs-id2497768\">Find the product:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>5<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>i<\/mi><\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>5<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>i<\/mi><\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1342702\" data-depth=\"2\">\n<h3 data-type=\"title\">Multiplying Complex Numbers Together<\/h3>\n<p id=\"fs-id2700341\">Now, let\u2019s multiply two complex numbers. We can use either the distributive property or more specifically the FOIL method because we are dealing with binomials. Recall that FOIL is an acronym for multiplying First, Inner, Outer, and Last terms together. The difference with complex numbers is that when we get a squared term,<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>i<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>i<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>it equals<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>\u22121.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>\u22121.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<div id=\"fs-id832128\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>c<\/mi><mo>+<\/mo><mi>d<\/mi><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mi>a<\/mi><mi>c<\/mi><mo>+<\/mo><mi>a<\/mi><mi>d<\/mi><mi>i<\/mi><mo>+<\/mo><mi>b<\/mi><mi>c<\/mi><mi>i<\/mi><mo>+<\/mo><mi>b<\/mi><mi>d<\/mi><msup>\n<mi>i<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mi>a<\/mi><mi>c<\/mi><mo>+<\/mo><mi>a<\/mi><mi>d<\/mi><mi>i<\/mi><mo>+<\/mo><mi>b<\/mi><mi>c<\/mi><mi>i<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>d<\/mi><\/mrow><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow><mspace width=\"2em\"><\/mspace>\n<msup>\n<mi>i<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>=<\/mo><mn>\u22121<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mi>c<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>d<\/mi><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mi>d<\/mi><mo>+<\/mo><mi>b<\/mi><mi>c<\/mi><mo stretchy=\"false\">)<\/mo><mi>i<\/mi><\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>Group\u00a0real\u00a0terms\u00a0and\u00a0imaginary\u00a0terms<\/mtext><mo>.<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>c<\/mi><mo>+<\/mo><mi>d<\/mi><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mi>a<\/mi><mi>c<\/mi><mo>+<\/mo><mi>a<\/mi><mi>d<\/mi><mi>i<\/mi><mo>+<\/mo><mi>b<\/mi><mi>c<\/mi><mi>i<\/mi><mo>+<\/mo><mi>b<\/mi><mi>d<\/mi><msup>\n<mi>i<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mi>a<\/mi><mi>c<\/mi><mo>+<\/mo><mi>a<\/mi><mi>d<\/mi><mi>i<\/mi><mo>+<\/mo><mi>b<\/mi><mi>c<\/mi><mi>i<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>d<\/mi><\/mrow><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow><mspace width=\"2em\"><\/mspace>\n<msup>\n<mi>i<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>=<\/mo><mn>\u22121<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mi>c<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>d<\/mi><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mi>d<\/mi><mo>+<\/mo><mi>b<\/mi><mi>c<\/mi><mo stretchy=\"false\">)<\/mo><mi>i<\/mi><\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>Group\u00a0real\u00a0terms\u00a0and\u00a0imaginary\u00a0terms<\/mtext><mo>.<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<div id=\"fs-id1758537\" class=\"how-to-notitle ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"How To\">\n<header>\n<h2 class=\"os-title\" data-type=\"title\" data-label-parent=\"How To\"><span class=\"os-title-label\">How To<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<p id=\"fs-id1457355\"><strong>Given two complex numbers, multiply to find the product.<\/strong><\/p>\n<ol id=\"fs-id1973809\" type=\"1\">\n<li>Use the distributive property or the FOIL method.<\/li>\n<li>Remember that<br \/>\n<math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>i<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>=<\/mo><mn>\u22121.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>i<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>=<\/mo><mn>\u22121.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\n<li>Group together the real terms and the imaginary terms<\/li>\n<\/ol>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"Example_02_04_05\" class=\"ui-has-child-title\" data-type=\"example\">\n<header>\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span><br \/>\n<span class=\"os-number\">5<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"body\">\n<div id=\"fs-id1311930\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id988839\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<h3 data-type=\"title\">Multiplying a Complex Number by a Complex Number<\/h3>\n<p id=\"fs-id1493312\">Multiply:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>4<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo>\u2212<\/mo><mn>5<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>4<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo>\u2212<\/mo><mn>5<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<details id=\"fs-id1435934\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\">\n<summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary>\n<section class=\"ui-body\" role=\"alert\">\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\n<div class=\"os-solution-container\">\n<div id=\"fs-id1813679\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo>\u2212<\/mo><mn>5<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>4<\/mn><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mn>4<\/mn><mo stretchy=\"false\">(<\/mo><mn>5<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mn>5<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mn>8<\/mn><mo>\u2212<\/mo><mn>20<\/mn><mi>i<\/mi><mo>+<\/mo><mn>6<\/mn><mi>i<\/mi><mo>\u2212<\/mo><mn>15<\/mn><mo>(<\/mo><msup>\n<mi>i<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>)<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mo stretchy=\"false\">(<\/mo><mn>8<\/mn><mo>+<\/mo><mn>15<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u221220<\/mn><mo>+<\/mo><mn>6<\/mn><mo stretchy=\"false\">)<\/mo><mi>i<\/mi><\/mrow><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>23<\/mn><mo>\u2212<\/mo><mn>14<\/mn><mi>i<\/mi><\/mrow><\/mtd>\n<\/mtr>\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo>\u2212<\/mo><mn>5<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>4<\/mn><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mn>4<\/mn><mo stretchy=\"false\">(<\/mo><mn>5<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mn>5<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mn>8<\/mn><mo>\u2212<\/mo><mn>20<\/mn><mi>i<\/mi><mo>+<\/mo><mn>6<\/mn><mi>i<\/mi><mo>\u2212<\/mo><mn>15<\/mn><mo>(<\/mo><msup>\n<mi>i<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>)<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mo stretchy=\"false\">(<\/mo><mn>8<\/mn><mo>+<\/mo><mn>15<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u221220<\/mn><mo>+<\/mo><mn>6<\/mn><mo stretchy=\"false\">)<\/mo><mi>i<\/mi><\/mrow><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>23<\/mn><mo>\u2212<\/mo><mn>14<\/mn><mi>i<\/mi><\/mrow><\/mtd>\n<\/mtr>\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<\/div>\n<\/section>\n<\/details>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2486744\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\">\n<header>\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span><br \/>\n<span class=\"os-number\">#5<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<div id=\"ti_02_04_05\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2507020\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"fs-id1475550\">Multiply:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>3<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi><\/mrow>\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\n<mrow>\n<mn>2<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi><\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>3<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi><\/mrow>\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\n<mrow>\n<mn>2<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi><\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/section>\n<section id=\"fs-id1555648\" data-depth=\"1\">\n<h2 data-type=\"title\">Dividing Complex Numbers<\/h2>\n<p id=\"fs-id1923471\">Dividing two complex numbers is more complicated than adding, subtracting, or multiplying because we cannot divide by an imaginary number, meaning that any fraction must have a real-number denominator to write the answer in standard form<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>We need to find a term by which we can multiply the numerator and the denominator that will eliminate the imaginary portion of the denominator so that we end up with a real number as the denominator. This term is called the complex conjugate of the denominator, which is found by changing the sign of the imaginary part of the complex number. In other words, the complex conjugate of<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>is<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>a<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>i<\/mi><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>a<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>i<\/mi><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>For example, the product of<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>a<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>i<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>a<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>i<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>is<\/p>\n<div id=\"fs-id781562\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\">\n<mrow>\n<msup>\n<mi>a<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mi>a<\/mi><mi>b<\/mi><mi>i<\/mi><mo>+<\/mo><mi>a<\/mi><mi>b<\/mi><mi>i<\/mi><mo>\u2212<\/mo><msup>\n<mi>b<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<msup>\n<mi>i<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<msup>\n<mi>a<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><msup>\n<mi>b<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\">\n<mrow>\n<msup>\n<mi>a<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mi>a<\/mi><mi>b<\/mi><mi>i<\/mi><mo>+<\/mo><mi>a<\/mi><mi>b<\/mi><mi>i<\/mi><mo>\u2212<\/mo><msup>\n<mi>b<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<msup>\n<mi>i<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<msup>\n<mi>a<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><msup>\n<mi>b<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1965413\">The result is a real number.<\/p>\n<p id=\"fs-id1466166\">Note that complex conjugates have an opposite relationship: The complex conjugate of<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>is<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>a<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>i<\/mi><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>a<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>i<\/mi><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>and the complex conjugate of<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>a<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>i<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>a<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>i<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>is<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>Further, when a quadratic equation with real coefficients has complex solutions, the solutions are always complex conjugates of one another.<\/p>\n<p id=\"fs-id2665321\">Suppose we want to divide<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>c<\/mi><mo>+<\/mo><mi>d<\/mi><mi>i<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>c<\/mi><mo>+<\/mo><mi>d<\/mi><mi>i<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>by<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>where neither<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>a<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>a<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>nor<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>b<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>b<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>equals zero. We first write the division as a fraction, then find the complex conjugate of the denominator, and multiply.<\/p>\n<div id=\"fs-id1539267\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mrow>\n<mi>c<\/mi><mo>+<\/mo><mi>d<\/mi><mi>i<\/mi>\n<\/mrow>\n<mrow>\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi>\n<\/mrow>\n<\/mfrac>\n<mspace width=\"0.5em\"><\/mspace><mtext>where\u00a0<\/mtext><mi>a<\/mi><mo>\u2260<\/mo><mn>0<\/mn><mspace width=\"0.5em\"><\/mspace><mtext>and\u00a0<\/mtext><mi>b<\/mi><mo>\u2260<\/mo><mn>0<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mrow>\n<mi>c<\/mi><mo>+<\/mo><mi>d<\/mi><mi>i<\/mi>\n<\/mrow>\n<mrow>\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi>\n<\/mrow>\n<\/mfrac>\n<mspace width=\"0.5em\"><\/mspace><mtext>where\u00a0<\/mtext><mi>a<\/mi><mo>\u2260<\/mo><mn>0<\/mn><mspace width=\"0.5em\"><\/mspace><mtext>and\u00a0<\/mtext><mi>b<\/mi><mo>\u2260<\/mo><mn>0<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id2504346\">Multiply the numerator and denominator by the complex conjugate of the denominator.<\/p>\n<div id=\"fs-id2015066\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>c<\/mi><mo>+<\/mo><mi>d<\/mi><mi>i<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<\/mfrac>\n<mo>\u22c5<\/mo><mfrac>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>a<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>i<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>a<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>i<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<\/mfrac>\n<mo>=<\/mo><mfrac>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>c<\/mi><mo>+<\/mo><mi>d<\/mi><mi>i<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\n<mrow>\n<mi>a<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>i<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\n<mrow>\n<mi>a<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>i<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>c<\/mi><mo>+<\/mo><mi>d<\/mi><mi>i<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<\/mfrac>\n<mo>\u22c5<\/mo><mfrac>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>a<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>i<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>a<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>i<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<\/mfrac>\n<mo>=<\/mo><mfrac>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>c<\/mi><mo>+<\/mo><mi>d<\/mi><mi>i<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\n<mrow>\n<mi>a<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>i<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\n<mrow>\n<mi>a<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>i<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<\/mfrac>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id2498465\">Apply the distributive property.<\/p>\n<div id=\"fs-id2497874\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mo>=<\/mo><mfrac>\n<mrow>\n<mi>c<\/mi><mi>a<\/mi><mo>\u2212<\/mo><mi>c<\/mi><mi>b<\/mi><mi>i<\/mi><mo>+<\/mo><mi>a<\/mi><mi>d<\/mi><mi>i<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>d<\/mi><msup>\n<mi>i<\/mi>\n<mn>2<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<mrow>\n<msup>\n<mi>a<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mi>a<\/mi><mi>b<\/mi><mi>i<\/mi><mo>+<\/mo><mi>a<\/mi><mi>b<\/mi><mi>i<\/mi><mo>\u2212<\/mo><msup>\n<mi>b<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<msup>\n<mi>i<\/mi>\n<mn>2<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mo>=<\/mo><mfrac>\n<mrow>\n<mi>c<\/mi><mi>a<\/mi><mo>\u2212<\/mo><mi>c<\/mi><mi>b<\/mi><mi>i<\/mi><mo>+<\/mo><mi>a<\/mi><mi>d<\/mi><mi>i<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>d<\/mi><msup>\n<mi>i<\/mi>\n<mn>2<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<mrow>\n<msup>\n<mi>a<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mi>a<\/mi><mi>b<\/mi><mi>i<\/mi><mo>+<\/mo><mi>a<\/mi><mi>b<\/mi><mi>i<\/mi><mo>\u2212<\/mo><msup>\n<mi>b<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<msup>\n<mi>i<\/mi>\n<mn>2<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mfrac>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1790276\">Simplify, remembering that<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>i<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>=<\/mo><mn>\u22121.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>i<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>=<\/mo><mn>\u22121.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<div id=\"fs-id2841772\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mo>=<\/mo><mfrac>\n<mrow>\n<mi>c<\/mi><mi>a<\/mi><mo>\u2212<\/mo><mi>c<\/mi><mi>b<\/mi><mi>i<\/mi><mo>+<\/mo><mi>a<\/mi><mi>d<\/mi><mi>i<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>d<\/mi><mo stretchy=\"false\">(<\/mo><mn>\u22121<\/mn><mo stretchy=\"false\">)<\/mo>\n<\/mrow>\n<mrow>\n<msup>\n<mi>a<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mi>a<\/mi><mi>b<\/mi><mi>i<\/mi><mo>+<\/mo><mi>a<\/mi><mi>b<\/mi><mi>i<\/mi><mo>\u2212<\/mo><msup>\n<mi>b<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo stretchy=\"false\">(<\/mo><mn>\u22121<\/mn><mo stretchy=\"false\">)<\/mo>\n<\/mrow>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mo>=<\/mo><mfrac>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mi>c<\/mi><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>d<\/mi><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mi>d<\/mi><mo>\u2212<\/mo><mi>c<\/mi><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><mi>i<\/mi>\n<\/mrow>\n<mrow>\n<msup>\n<mi>a<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><msup>\n<mi>b<\/mi>\n<mn>2<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mo>=<\/mo><mfrac>\n<mrow>\n<mi>c<\/mi><mi>a<\/mi><mo>\u2212<\/mo><mi>c<\/mi><mi>b<\/mi><mi>i<\/mi><mo>+<\/mo><mi>a<\/mi><mi>d<\/mi><mi>i<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>d<\/mi><mo stretchy=\"false\">(<\/mo><mn>\u22121<\/mn><mo stretchy=\"false\">)<\/mo>\n<\/mrow>\n<mrow>\n<msup>\n<mi>a<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mi>a<\/mi><mi>b<\/mi><mi>i<\/mi><mo>+<\/mo><mi>a<\/mi><mi>b<\/mi><mi>i<\/mi><mo>\u2212<\/mo><msup>\n<mi>b<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo stretchy=\"false\">(<\/mo><mn>\u22121<\/mn><mo stretchy=\"false\">)<\/mo>\n<\/mrow>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mo>=<\/mo><mfrac>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mi>c<\/mi><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>d<\/mi><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mi>d<\/mi><mo>\u2212<\/mo><mi>c<\/mi><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><mi>i<\/mi>\n<\/mrow>\n<mrow>\n<msup>\n<mi>a<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><msup>\n<mi>b<\/mi>\n<mn>2<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<div id=\"fs-id1392562\" class=\"ui-has-child-title\" data-type=\"note\">\n<header>\n<h2 class=\"os-title\" data-type=\"title\"><span class=\"os-title-label\" data-type=\"\">The Complex Conjugate<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<p id=\"fs-id1280042\">The <span id=\"term-00005\" data-type=\"term\">complex conjugate<\/span> of a complex number<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>is<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>a<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>i<\/mi><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>a<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>i<\/mi><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>It is found by changing the sign of the imaginary part of the complex number. The real part of the number is left unchanged.<\/p>\n<ul id=\"fs-id1435532\">\n<li>When a complex number is multiplied by its complex conjugate, the result is a real number.<\/li>\n<li>When a complex number is added to its complex conjugate, the result is a real number.<\/li>\n<\/ul>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"Example_02_04_06\" class=\"ui-has-child-title\" data-type=\"example\">\n<header>\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span><br \/>\n<span class=\"os-number\">6<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"body\">\n<div id=\"fs-id1332427\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1208193\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<h3 data-type=\"title\">Finding Complex Conjugates<\/h3>\n<p id=\"fs-id791546\">Find the complex conjugate of each number.<\/p>\n<ol id=\"fs-id1183222\" class=\"circled\" type=\"1\">\n<li><span class=\"token\">\u24d0<\/span><br \/>\n<math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>2<\/mn><mo>+<\/mo><mi>i<\/mi><msqrt>\n<mn>5<\/mn>\n<\/msqrt>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>2<\/mn><mo>+<\/mo><mi>i<\/mi><msqrt>\n<mn>5<\/mn>\n<\/msqrt>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\n<li><span class=\"token\">\u24d1<\/span><br \/>\n<math display=\"inline\"><semantics><mrow>\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<mi>i<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<mi>i<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\n<\/ol>\n<\/div>\n<\/div>\n<details id=\"fs-id2935900\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\">\n<summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary>\n<section class=\"ui-body\" role=\"alert\">\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\n<div class=\"os-solution-container\">\n<ol id=\"fs-id1216223\" class=\"circled\" type=\"1\">\n<li><span class=\"token\">\u24d0<\/span>The number is already in the form<br \/>\n<math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>The complex conjugate is<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>a<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>i<\/mi><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>a<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>i<\/mi><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>or<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>2<\/mn><mo>\u2212<\/mo><mi>i<\/mi><msqrt>\n<mn>5<\/mn>\n<\/msqrt>\n<mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>2<\/mn><mo>\u2212<\/mo><mi>i<\/mi><msqrt>\n<mn>5<\/mn>\n<\/msqrt>\n<mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\n<li><span class=\"token\">\u24d1<\/span>We can rewrite this number in the form<br \/>\n<math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>i<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>as<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>0<\/mn><mo>\u2212<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<mi>i<\/mi><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>0<\/mn><mo>\u2212<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<mi>i<\/mi><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>The complex conjugate is<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>a<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>i<\/mi><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>a<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>i<\/mi><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>or<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>0<\/mn><mo>+<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<mi>i<\/mi><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>0<\/mn><mo>+<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<mi>i<\/mi><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>This can be written simply as<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<mi>i<\/mi><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<mi>i<\/mi><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\n<\/ol>\n<\/div>\n<\/section>\n<\/details>\n<div id=\"fs-id1425864\" data-type=\"commentary\">\n<h3 data-type=\"commentary-title\"><span class=\"os-title-label\">Analysis <\/span><\/h3>\n<p id=\"fs-id2315664\">Although we have seen that we can find the complex conjugate of an imaginary number, in practice we generally find the complex conjugates of only complex numbers with both a real and an imaginary component. To obtain a real number from an imaginary number, we can simply multiply by<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>i<\/mi><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>i<\/mi><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1445609\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\">\n<header>\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span><br \/>\n<span class=\"os-number\">#6<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<div id=\"ti_02_04_06\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1562514\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"fs-id1562515\">Find the complex conjugate of<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>\u22123<\/mn><mo>+<\/mo><mn>4<\/mn><mi>i<\/mi><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>\u22123<\/mn><mo>+<\/mo><mn>4<\/mn><mi>i<\/mi><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2521760\" class=\"how-to-notitle ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"How To\">\n<header>\n<h2 class=\"os-title\" data-type=\"title\" data-label-parent=\"How To\"><span class=\"os-title-label\">How To<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<p id=\"fs-id1395747\"><strong>Given two complex numbers, divide one by the other.<\/strong><\/p>\n<ol id=\"fs-id2753766\" type=\"1\">\n<li>Write the division problem as a fraction.<\/li>\n<li>Determine the complex conjugate of the denominator.<\/li>\n<li>Multiply the numerator and denominator of the fraction by the complex conjugate of the denominator.<\/li>\n<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"Example_02_04_07\" class=\"ui-has-child-title\" data-type=\"example\">\n<header>\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span><br \/>\n<span class=\"os-number\">7<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"body\">\n<div id=\"fs-id1462292\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1883764\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<h3 data-type=\"title\">Dividing Complex Numbers<\/h3>\n<p id=\"fs-id1531602\">Divide:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>2<\/mn><mo>+<\/mo><mn>5<\/mn><mi>i<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>2<\/mn><mo>+<\/mo><mn>5<\/mn><mi>i<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow><\/annotation-xml><\/semantics><\/math>by<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>4<\/mn><mo>\u2212<\/mo><mi>i<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>4<\/mn><mo>\u2212<\/mo><mi>i<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<details id=\"fs-id1511138\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\">\n<summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary>\n<section class=\"ui-body\" role=\"alert\">\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\n<div class=\"os-solution-container\">\n<p id=\"fs-id2483518\">We begin by writing the problem as a fraction.<\/p>\n<div id=\"fs-id1736666\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>2<\/mn><mo>+<\/mo><mn>5<\/mn><mi>i<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>4<\/mn><mo>\u2212<\/mo><mi>i<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>2<\/mn><mo>+<\/mo><mn>5<\/mn><mi>i<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>4<\/mn><mo>\u2212<\/mo><mi>i<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<\/mfrac>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id3119235\">Then we multiply the numerator and denominator by the complex conjugate of the denominator.<\/p>\n<div id=\"fs-id1500527\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo>+<\/mo><mn>5<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo>\n<\/mrow>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo>\u2212<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo>\n<\/mrow>\n<\/mfrac>\n<mspace width=\"0.5em\"><\/mspace><mo>\u22c5<\/mo><mfrac>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo>+<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo>\n<\/mrow>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo>+<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo>\n<\/mrow>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo>+<\/mo><mn>5<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo>\n<\/mrow>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo>\u2212<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo>\n<\/mrow>\n<\/mfrac>\n<mspace width=\"0.5em\"><\/mspace><mo>\u22c5<\/mo><mfrac>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo>+<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo>\n<\/mrow>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo>+<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo>\n<\/mrow>\n<\/mfrac>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1204485\">To multiply two complex numbers, we expand the product as we would with polynomials (using FOIL).<\/p>\n<div id=\"fs-id2906059\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow>\n<mfrac>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo>+<\/mo><mn>5<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo>\u2212<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mfrac>\n<mo>\u22c5<\/mo><mfrac>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo>+<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo>+<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mfrac>\n<mrow>\n<mn>8<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi><mo>+<\/mo><mn>20<\/mn><mi>i<\/mi><mo>+<\/mo><mn>5<\/mn><msup>\n<mi>i<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<\/mrow>\n<mrow>\n<mn>16<\/mn><mo>+<\/mo><mn>4<\/mn><mi>i<\/mi><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi><mo>\u2212<\/mo><msup>\n<mi>i<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<\/mrow>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mfrac>\n<mrow>\n<mn>8<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi><mo>+<\/mo><mn>20<\/mn><mi>i<\/mi><mo>+<\/mo><mn>5<\/mn><mo stretchy=\"false\">(<\/mo><mn>\u22121<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<mrow>\n<mn>16<\/mn><mo>+<\/mo><mn>4<\/mn><mi>i<\/mi><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u22121<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>Because<\/mtext><mo>\u00a0<\/mo><msup>\n<mi>i<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>=<\/mo><mn>\u22121.<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mfrac>\n<mrow>\n<mn>3<\/mn><mo>+<\/mo><mn>22<\/mn><mi>i<\/mi><\/mrow>\n<mrow>\n<mn>17<\/mn><\/mrow>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mfrac>\n<mn>3<\/mn>\n<mrow>\n<mn>17<\/mn><\/mrow>\n<\/mfrac>\n<mo>+<\/mo><mfrac>\n<mrow>\n<mn>22<\/mn><\/mrow>\n<mrow>\n<mn>17<\/mn><\/mrow>\n<\/mfrac>\n<mi>i<\/mi><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>Separate\u00a0real\u00a0and\u00a0imaginary\u00a0parts<\/mtext><mo>.<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\"><mrow>\n<mfrac>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo>+<\/mo><mn>5<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo>\u2212<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mfrac>\n<mo>\u22c5<\/mo><mfrac>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo>+<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo>+<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mfrac>\n<mrow>\n<mn>8<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi><mo>+<\/mo><mn>20<\/mn><mi>i<\/mi><mo>+<\/mo><mn>5<\/mn><msup>\n<mi>i<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<\/mrow>\n<mrow>\n<mn>16<\/mn><mo>+<\/mo><mn>4<\/mn><mi>i<\/mi><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi><mo>\u2212<\/mo><msup>\n<mi>i<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<\/mrow>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mfrac>\n<mrow>\n<mn>8<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi><mo>+<\/mo><mn>20<\/mn><mi>i<\/mi><mo>+<\/mo><mn>5<\/mn><mo stretchy=\"false\">(<\/mo><mn>\u22121<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<mrow>\n<mn>16<\/mn><mo>+<\/mo><mn>4<\/mn><mi>i<\/mi><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u22121<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>Because<\/mtext><mo>\u00a0<\/mo><msup>\n<mi>i<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>=<\/mo><mn>\u22121.<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mfrac>\n<mrow>\n<mn>3<\/mn><mo>+<\/mo><mn>22<\/mn><mi>i<\/mi><\/mrow>\n<mrow>\n<mn>17<\/mn><\/mrow>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mfrac>\n<mn>3<\/mn>\n<mrow>\n<mn>17<\/mn><\/mrow>\n<\/mfrac>\n<mo>+<\/mo><mfrac>\n<mrow>\n<mn>22<\/mn><\/mrow>\n<mrow>\n<mn>17<\/mn><\/mrow>\n<\/mfrac>\n<mi>i<\/mi><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>Separate\u00a0real\u00a0and\u00a0imaginary\u00a0parts<\/mtext><mo>.<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id2486676\">Note that this expresses the quotient in standard form.<\/p>\n<\/div>\n<\/section>\n<\/details>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1555717\" data-depth=\"1\">\n<h2 data-type=\"title\">Simplifying Powers of <em data-effect=\"italics\">i<\/em><\/h2>\n<p id=\"fs-id2049707\">The powers of<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>i<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>i<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>are cyclic. Let\u2019s look at what happens when we raise<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>i<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>i<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>to increasing powers.<\/p>\n<div id=\"fs-id1537277\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<msup>\n<mi>i<\/mi>\n<mn>1<\/mn>\n<\/msup>\n<mo>=<\/mo><mi>i<\/mi>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<msup>\n<mi>i<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>=<\/mo><mn>\u22121<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<msup>\n<mi>i<\/mi>\n<mn>3<\/mn>\n<\/msup>\n<mo>=<\/mo><msup>\n<mi>i<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u22c5<\/mo><mi>i<\/mi><mo>=<\/mo><mn>\u22121<\/mn><mo>\u22c5<\/mo><mi>i<\/mi><mo>=<\/mo><mi>\u2212<\/mi><mi>i<\/mi>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<msup>\n<mi>i<\/mi>\n<mn>4<\/mn>\n<\/msup>\n<mo>=<\/mo><msup>\n<mi>i<\/mi>\n<mn>3<\/mn>\n<\/msup>\n<mo>\u22c5<\/mo><mi>i<\/mi><mo>=<\/mo><mi>\u2212<\/mi><mi>i<\/mi><mo>\u22c5<\/mo><mi>i<\/mi><mo>=<\/mo><mi>\u2212<\/mi><msup>\n<mi>i<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>=<\/mo><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u22121<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>1<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<msup>\n<mi>i<\/mi>\n<mn>5<\/mn>\n<\/msup>\n<mo>=<\/mo><msup>\n<mi>i<\/mi>\n<mn>4<\/mn>\n<\/msup>\n<mo>\u22c5<\/mo><mi>i<\/mi><mo>=<\/mo><mn>1<\/mn><mo>\u22c5<\/mo><mi>i<\/mi><mo>=<\/mo><mi>i<\/mi>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<msup>\n<mi>i<\/mi>\n<mn>1<\/mn>\n<\/msup>\n<mo>=<\/mo><mi>i<\/mi>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<msup>\n<mi>i<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>=<\/mo><mn>\u22121<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<msup>\n<mi>i<\/mi>\n<mn>3<\/mn>\n<\/msup>\n<mo>=<\/mo><msup>\n<mi>i<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u22c5<\/mo><mi>i<\/mi><mo>=<\/mo><mn>\u22121<\/mn><mo>\u22c5<\/mo><mi>i<\/mi><mo>=<\/mo><mi>\u2212<\/mi><mi>i<\/mi>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<msup>\n<mi>i<\/mi>\n<mn>4<\/mn>\n<\/msup>\n<mo>=<\/mo><msup>\n<mi>i<\/mi>\n<mn>3<\/mn>\n<\/msup>\n<mo>\u22c5<\/mo><mi>i<\/mi><mo>=<\/mo><mi>\u2212<\/mi><mi>i<\/mi><mo>\u22c5<\/mo><mi>i<\/mi><mo>=<\/mo><mi>\u2212<\/mi><msup>\n<mi>i<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>=<\/mo><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u22121<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>1<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<msup>\n<mi>i<\/mi>\n<mn>5<\/mn>\n<\/msup>\n<mo>=<\/mo><msup>\n<mi>i<\/mi>\n<mn>4<\/mn>\n<\/msup>\n<mo>\u22c5<\/mo><mi>i<\/mi><mo>=<\/mo><mn>1<\/mn><mo>\u22c5<\/mo><mi>i<\/mi><mo>=<\/mo><mi>i<\/mi>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id2498837\">We can see that when we get to the fifth power of<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>i<\/mi><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>i<\/mi><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>it is equal to the first power. As we continue to multiply<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>i<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>i<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>by increasing powers, we will see a cycle of four. Let\u2019s examine the next four powers of<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>i<\/mi><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>i<\/mi><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<div id=\"fs-id1943372\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<msup>\n<mi>i<\/mi>\n<mn>6<\/mn>\n<\/msup>\n<mo>=<\/mo><msup>\n<mi>i<\/mi>\n<mn>5<\/mn>\n<\/msup>\n<mo>\u22c5<\/mo><mi>i<\/mi><mo>=<\/mo><mi>i<\/mi><mo>\u22c5<\/mo><mi>i<\/mi><mo>=<\/mo><msup>\n<mi>i<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>=<\/mo><mn>\u22121<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<msup>\n<mi>i<\/mi>\n<mn>7<\/mn>\n<\/msup>\n<mo>=<\/mo><msup>\n<mi>i<\/mi>\n<mn>6<\/mn>\n<\/msup>\n<mo>\u22c5<\/mo><mi>i<\/mi><mo>=<\/mo><msup>\n<mi>i<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u22c5<\/mo><mi>i<\/mi><mo>=<\/mo><msup>\n<mi>i<\/mi>\n<mn>3<\/mn>\n<\/msup>\n<mo>=<\/mo><mi>\u2212<\/mi><mi>i<\/mi>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<msup>\n<mi>i<\/mi>\n<mn>8<\/mn>\n<\/msup>\n<mo>=<\/mo><msup>\n<mi>i<\/mi>\n<mn>7<\/mn>\n<\/msup>\n<mo>\u22c5<\/mo><mi>i<\/mi><mo>=<\/mo><msup>\n<mi>i<\/mi>\n<mn>3<\/mn>\n<\/msup>\n<mo>\u22c5<\/mo><mi>i<\/mi><mo>=<\/mo><msup>\n<mi>i<\/mi>\n<mn>4<\/mn>\n<\/msup>\n<mo>=<\/mo><mn>1<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<msup>\n<mi>i<\/mi>\n<mn>9<\/mn>\n<\/msup>\n<mo>=<\/mo><msup>\n<mi>i<\/mi>\n<mn>8<\/mn>\n<\/msup>\n<mo>\u22c5<\/mo><mi>i<\/mi><mo>=<\/mo><msup>\n<mi>i<\/mi>\n<mn>4<\/mn>\n<\/msup>\n<mo>\u22c5<\/mo><mi>i<\/mi><mo>=<\/mo><msup>\n<mi>i<\/mi>\n<mn>5<\/mn>\n<\/msup>\n<mo>=<\/mo><mi>i<\/mi>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<msup>\n<mi>i<\/mi>\n<mn>6<\/mn>\n<\/msup>\n<mo>=<\/mo><msup>\n<mi>i<\/mi>\n<mn>5<\/mn>\n<\/msup>\n<mo>\u22c5<\/mo><mi>i<\/mi><mo>=<\/mo><mi>i<\/mi><mo>\u22c5<\/mo><mi>i<\/mi><mo>=<\/mo><msup>\n<mi>i<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>=<\/mo><mn>\u22121<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<msup>\n<mi>i<\/mi>\n<mn>7<\/mn>\n<\/msup>\n<mo>=<\/mo><msup>\n<mi>i<\/mi>\n<mn>6<\/mn>\n<\/msup>\n<mo>\u22c5<\/mo><mi>i<\/mi><mo>=<\/mo><msup>\n<mi>i<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u22c5<\/mo><mi>i<\/mi><mo>=<\/mo><msup>\n<mi>i<\/mi>\n<mn>3<\/mn>\n<\/msup>\n<mo>=<\/mo><mi>\u2212<\/mi><mi>i<\/mi>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<msup>\n<mi>i<\/mi>\n<mn>8<\/mn>\n<\/msup>\n<mo>=<\/mo><msup>\n<mi>i<\/mi>\n<mn>7<\/mn>\n<\/msup>\n<mo>\u22c5<\/mo><mi>i<\/mi><mo>=<\/mo><msup>\n<mi>i<\/mi>\n<mn>3<\/mn>\n<\/msup>\n<mo>\u22c5<\/mo><mi>i<\/mi><mo>=<\/mo><msup>\n<mi>i<\/mi>\n<mn>4<\/mn>\n<\/msup>\n<mo>=<\/mo><mn>1<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<msup>\n<mi>i<\/mi>\n<mn>9<\/mn>\n<\/msup>\n<mo>=<\/mo><msup>\n<mi>i<\/mi>\n<mn>8<\/mn>\n<\/msup>\n<mo>\u22c5<\/mo><mi>i<\/mi><mo>=<\/mo><msup>\n<mi>i<\/mi>\n<mn>4<\/mn>\n<\/msup>\n<mo>\u22c5<\/mo><mi>i<\/mi><mo>=<\/mo><msup>\n<mi>i<\/mi>\n<mn>5<\/mn>\n<\/msup>\n<mo>=<\/mo><mi>i<\/mi>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1442394\">The cycle is repeated continuously:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>i<\/mi><mo>,<\/mo><mn>\u22121<\/mn><mo>,<\/mo><mo>\u2212<\/mo><mi>i<\/mi><mo>,<\/mo><mn>1<\/mn><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>i<\/mi><mo>,<\/mo><mn>\u22121<\/mn><mo>,<\/mo><mo>\u2212<\/mo><mi>i<\/mi><mo>,<\/mo><mn>1<\/mn><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>every four powers.<\/p>\n<div id=\"Example_02_04_08\" class=\"ui-has-child-title\" data-type=\"example\">\n<header>\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span><br \/>\n<span class=\"os-number\">8<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"body\">\n<div id=\"fs-id1831392\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2440261\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<h3 data-type=\"title\">Simplifying Powers of <math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>i<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>i<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/h3>\n<p id=\"fs-id1716914\">Evaluate:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>i<\/mi>\n<mrow>\n<mn>35<\/mn>\n<\/mrow>\n<\/msup>\n<mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>i<\/mi>\n<mrow>\n<mn>35<\/mn>\n<\/mrow>\n<\/msup>\n<mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<details id=\"fs-id2435624\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\">\n<summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary>\n<section class=\"ui-body\" role=\"alert\">\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\n<div class=\"os-solution-container\">\n<p id=\"fs-id1269680\">Since<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>i<\/mi>\n<mn>4<\/mn>\n<\/msup>\n<mo>=<\/mo><mn>1<\/mn><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>i<\/mi>\n<mn>4<\/mn>\n<\/msup>\n<mo>=<\/mo><mn>1<\/mn><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>we can simplify the problem by factoring out as many factors of<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>i<\/mi>\n<mn>4<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>i<\/mi>\n<mn>4<\/mn>\n<\/msup>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math>as possible. To do so, first determine how many times 4 goes into 35:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>35<\/mn><mo>=<\/mo><mn>4<\/mn><mo>\u22c5<\/mo><mn>8<\/mn><mo>+<\/mo><mn>3.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>35<\/mn><mo>=<\/mo><mn>4<\/mn><mo>\u22c5<\/mo><mn>8<\/mn><mo>+<\/mo><mn>3.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<div id=\"fs-id1257595\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>i<\/mi>\n<mrow>\n<mn>35<\/mn>\n<\/mrow>\n<\/msup>\n<mo>=<\/mo><msup>\n<mi>i<\/mi>\n<mrow>\n<mn>4<\/mn><mo>\u22c5<\/mo><mn>8<\/mn><mo>+<\/mo><mn>3<\/mn>\n<\/mrow>\n<\/msup>\n<mo>=<\/mo><msup>\n<mi>i<\/mi>\n<mrow>\n<mn>4<\/mn><mo>\u22c5<\/mo><mn>8<\/mn>\n<\/mrow>\n<\/msup>\n<mo>\u22c5<\/mo><msup>\n<mi>i<\/mi>\n<mn>3<\/mn>\n<\/msup>\n<mo>=<\/mo><msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<msup>\n<mi>i<\/mi>\n<mn>4<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<mn>8<\/mn>\n<\/msup>\n<mo>\u22c5<\/mo><msup>\n<mi>i<\/mi>\n<mn>3<\/mn>\n<\/msup>\n<mo>=<\/mo><msup>\n<mn>1<\/mn>\n<mn>8<\/mn>\n<\/msup>\n<mo>\u22c5<\/mo><msup>\n<mi>i<\/mi>\n<mn>3<\/mn>\n<\/msup>\n<mo>=<\/mo><msup>\n<mi>i<\/mi>\n<mn>3<\/mn>\n<\/msup>\n<mo>=<\/mo><mo>\u2212<\/mo><mi>i<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>i<\/mi>\n<mrow>\n<mn>35<\/mn>\n<\/mrow>\n<\/msup>\n<mo>=<\/mo><msup>\n<mi>i<\/mi>\n<mrow>\n<mn>4<\/mn><mo>\u22c5<\/mo><mn>8<\/mn><mo>+<\/mo><mn>3<\/mn>\n<\/mrow>\n<\/msup>\n<mo>=<\/mo><msup>\n<mi>i<\/mi>\n<mrow>\n<mn>4<\/mn><mo>\u22c5<\/mo><mn>8<\/mn>\n<\/mrow>\n<\/msup>\n<mo>\u22c5<\/mo><msup>\n<mi>i<\/mi>\n<mn>3<\/mn>\n<\/msup>\n<mo>=<\/mo><msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<msup>\n<mi>i<\/mi>\n<mn>4<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<mn>8<\/mn>\n<\/msup>\n<mo>\u22c5<\/mo><msup>\n<mi>i<\/mi>\n<mn>3<\/mn>\n<\/msup>\n<mo>=<\/mo><msup>\n<mn>1<\/mn>\n<mn>8<\/mn>\n<\/msup>\n<mo>\u22c5<\/mo><msup>\n<mi>i<\/mi>\n<mn>3<\/mn>\n<\/msup>\n<mo>=<\/mo><msup>\n<mi>i<\/mi>\n<mn>3<\/mn>\n<\/msup>\n<mo>=<\/mo><mo>\u2212<\/mo><mi>i<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<\/div>\n<\/section>\n<\/details>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1845345\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\">\n<header>\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span><br \/>\n<span class=\"os-number\">#7<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<div id=\"ti_02_04_07\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1745095\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"fs-id1745096\">Evaluate:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>i<\/mi>\n<mrow>\n<mn>18<\/mn>\n<\/mrow>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>i<\/mi>\n<mrow>\n<mn>18<\/mn>\n<\/mrow>\n<\/msup>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1567766\" class=\"precalculus qa ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Q&amp;A\">\n<header>\n<h2 class=\"os-title\" data-type=\"title\" data-label-parent=\"Q&amp;A\"><span class=\"os-title-label\">Q&amp;A<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<p id=\"fs-id2683377\"><strong>Can we write <\/strong><\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>i<\/mi>\n<mrow>\n<mn>35<\/mn>\n<\/mrow>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>i<\/mi>\n<mrow>\n<mn>35<\/mn>\n<\/mrow>\n<\/msup>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math>in other helpful ways?<\/p>\n<p id=\"fs-id1557516\"><em data-effect=\"italics\">As we saw in <a class=\"autogenerated-content\" href=\"2-4-complex-numbers#Example_02_04_08\">Example 8<\/a>, we reduced <\/em><\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>i<\/mi>\n<mrow>\n<mn>35<\/mn>\n<\/mrow>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>i<\/mi>\n<mrow>\n<mn>35<\/mn>\n<\/mrow>\n<\/msup>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math>to<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>i<\/mi>\n<mn>3<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>i<\/mi>\n<mn>3<\/mn>\n<\/msup>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math>by dividing the exponent by 4 and using the remainder to find the simplified form. But perhaps another factorization of<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>i<\/mi>\n<mrow>\n<mn>35<\/mn>\n<\/mrow>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>i<\/mi>\n<mrow>\n<mn>35<\/mn>\n<\/mrow>\n<\/msup>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math>may be more useful. <a class=\"autogenerated-content\" href=\"2-4-complex-numbers#Table_02_04_01\">Table 1<\/a> shows some other possible factorizations.<\/p>\n<div id=\"Table_02_04_01\" class=\"os-table\">\n<table data-id=\"Table_02_04_01\">\n<tbody>\n<tr>\n<td data-align=\"center\"><strong>Factorization of<\/strong><\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>i<\/mi>\n<mrow>\n<mn>35<\/mn>\n<\/mrow>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>i<\/mi>\n<mrow>\n<mn>35<\/mn>\n<\/mrow>\n<\/msup>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>i<\/mi>\n<mrow>\n<mn>34<\/mn>\n<\/mrow>\n<\/msup>\n<mo>\u22c5<\/mo><mi>i<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>i<\/mi>\n<mrow>\n<mn>34<\/mn>\n<\/mrow>\n<\/msup>\n<mo>\u22c5<\/mo><mi>i<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>i<\/mi>\n<mrow>\n<mn>33<\/mn>\n<\/mrow>\n<\/msup>\n<mo>\u22c5<\/mo><msup>\n<mi>i<\/mi>\n<mn>2<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>i<\/mi>\n<mrow>\n<mn>33<\/mn>\n<\/mrow>\n<\/msup>\n<mo>\u22c5<\/mo><msup>\n<mi>i<\/mi>\n<mn>2<\/mn>\n<\/msup>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>i<\/mi>\n<mrow>\n<mn>31<\/mn>\n<\/mrow>\n<\/msup>\n<mo>\u22c5<\/mo><msup>\n<mi>i<\/mi>\n<mn>4<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>i<\/mi>\n<mrow>\n<mn>31<\/mn>\n<\/mrow>\n<\/msup>\n<mo>\u22c5<\/mo><msup>\n<mi>i<\/mi>\n<mn>4<\/mn>\n<\/msup>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>i<\/mi>\n<mrow>\n<mn>19<\/mn>\n<\/mrow>\n<\/msup>\n<mo>\u22c5<\/mo><msup>\n<mi>i<\/mi>\n<mrow>\n<mn>16<\/mn>\n<\/mrow>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>i<\/mi>\n<mrow>\n<mn>19<\/mn>\n<\/mrow>\n<\/msup>\n<mo>\u22c5<\/mo><msup>\n<mi>i<\/mi>\n<mrow>\n<mn>16<\/mn>\n<\/mrow>\n<\/msup>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\"><strong>Reduced form<\/strong><\/td>\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<msup>\n<mi>i<\/mi>\n<mn>2<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<mrow>\n<mn>17<\/mn>\n<\/mrow>\n<\/msup>\n<mo>\u22c5<\/mo><mi>i<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<msup>\n<mi>i<\/mi>\n<mn>2<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<mrow>\n<mn>17<\/mn>\n<\/mrow>\n<\/msup>\n<mo>\u22c5<\/mo><mi>i<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>i<\/mi>\n<mrow>\n<mn>33<\/mn>\n<\/mrow>\n<\/msup>\n<mo>\u22c5<\/mo><mrow><mo>(<\/mo>\n<mrow>\n<mn>\u22121<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>i<\/mi>\n<mrow>\n<mn>33<\/mn>\n<\/mrow>\n<\/msup>\n<mo>\u22c5<\/mo><mrow><mo>(<\/mo>\n<mrow>\n<mn>\u22121<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>i<\/mi>\n<mrow>\n<mn>31<\/mn>\n<\/mrow>\n<\/msup>\n<mo>\u22c5<\/mo><mn>1<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>i<\/mi>\n<mrow>\n<mn>31<\/mn>\n<\/mrow>\n<\/msup>\n<mo>\u22c5<\/mo><mn>1<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\n<td data-align=\"center\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>i<\/mi>\n<mrow>\n<mn>19<\/mn>\n<\/mrow>\n<\/msup>\n<mo>\u22c5<\/mo><msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<msup>\n<mi>i<\/mi>\n<mn>4<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<mn>4<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>i<\/mi>\n<mrow>\n<mn>19<\/mn>\n<\/mrow>\n<\/msup>\n<mo>\u22c5<\/mo><msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<msup>\n<mi>i<\/mi>\n<mn>4<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<mn>4<\/mn>\n<\/msup>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\"><strong>Simplified form<\/strong><\/td>\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>\u22121<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<mrow>\n<mn>17<\/mn>\n<\/mrow>\n<\/msup>\n<mo>\u22c5<\/mo><mi>i<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>\u22121<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<mrow>\n<mn>17<\/mn>\n<\/mrow>\n<\/msup>\n<mo>\u22c5<\/mo><mi>i<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mo>\u2212<\/mo><msup>\n<mi>i<\/mi>\n<mrow>\n<mn>33<\/mn>\n<\/mrow>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mo>\u2212<\/mo><msup>\n<mi>i<\/mi>\n<mrow>\n<mn>33<\/mn>\n<\/mrow>\n<\/msup>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>i<\/mi>\n<mrow>\n<mn>31<\/mn>\n<\/mrow>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>i<\/mi>\n<mrow>\n<mn>31<\/mn>\n<\/mrow>\n<\/msup>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>i<\/mi>\n<mrow>\n<mn>19<\/mn>\n<\/mrow>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>i<\/mi>\n<mrow>\n<mn>19<\/mn>\n<\/mrow>\n<\/msup>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Table <\/span><br \/>\n<span class=\"os-number\">1<\/span><\/p>\n<p><span class=\"os-caption\"><span class=\"os-caption\"><br \/>\n<\/span><\/span>&nbsp;<\/p>\n<p><span class=\"os-caption\"><br \/>\n<\/span><\/div>\n<\/div>\n<p id=\"fs-id1164415\"><em data-effect=\"italics\">Each of these will eventually result in the answer we obtained above but may require several more steps than our earlier method.<\/em><\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2888800\" class=\"media-notitle ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Media\">\n<header>\n<h2 class=\"os-title\" data-type=\"title\" data-label-parent=\"Media\"><span class=\"os-title-label\">Media<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<p id=\"fs-id1339843\">Access these online resources for additional instruction and practice with complex numbers.<\/p>\n<ul id=\"fs-id2876017\">\n<li><a href=\"http:\/\/openstax.org\/l\/addsubcomplex\" target=\"_blank\" rel=\"noopener nofollow\">Adding and Subtracting Complex Numbers<\/a><\/li>\n<li><a href=\"http:\/\/openstax.org\/l\/multiplycomplex\" target=\"_blank\" rel=\"noopener nofollow\">Multiply Complex Numbers<\/a><\/li>\n<li><a href=\"http:\/\/openstax.org\/l\/multcompconj\" target=\"_blank\" rel=\"noopener nofollow\">Multiplying Complex Conjugates<\/a><\/li>\n<li><a href=\"http:\/\/openstax.org\/l\/raisingi\" target=\"_blank\" rel=\"noopener nofollow\">Raising <em data-effect=\"italics\">i<\/em> to Powers<\/a><\/li>\n<\/ul>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<div class=\"os-eos os-section-exercises-container\" data-uuid-key=\".section-exercises\">\n<h2 data-type=\"document-title\" data-rex-keep=\"true\"><span class=\"os-text\">2.4 Section Exercises<\/span><\/h2>\n<section id=\"fs-id3236190\" class=\"section-exercises\" data-depth=\"1\">\n<section id=\"fs-id1399221\" data-depth=\"2\">\n<h3 data-type=\"title\">Verbal<\/h3>\n<div id=\"fs-id1418488\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id767934\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1418488-solution\">1<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id767935\">Explain how to add complex numbers.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2638642\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1561233\" data-type=\"problem\"><span class=\"os-number\">2<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1561234\">What is the basic principle in multiplication of complex numbers?<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1265940\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1265942\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1265940-solution\">3<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1181910\">Give an example to show that the product of two imaginary numbers is not always imaginary.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1289893\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1289894\" data-type=\"problem\"><span class=\"os-number\">4<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1289895\">What is a characteristic of the plot of a real number in the complex plane?<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id2979908\" data-depth=\"2\">\n<h3 data-type=\"title\">Algebraic<\/h3>\n<p id=\"fs-id1722892\">For the following exercises, evaluate the algebraic expressions.<\/p>\n<div id=\"fs-id1295468\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2906264\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1295468-solution\">5<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2906265\">If<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>y<\/mi><mo>=<\/mo><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><mo>,<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>y<\/mi><mo>=<\/mo><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><mo>,<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>evaluate<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>y<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>y<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>given<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>x<\/mi><mo>=<\/mo><mn>2<\/mn><mi>i<\/mi><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>x<\/mi><mo>=<\/mo><mn>2<\/mn><mi>i<\/mi><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1332284\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1332285\" data-type=\"problem\"><span class=\"os-number\">6<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id3229108\">If<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>y<\/mi><mo>=<\/mo><msup>\n<mi>x<\/mi>\n<mn>3<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>2<\/mn><mo>,<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>y<\/mi><mo>=<\/mo><msup>\n<mi>x<\/mi>\n<mn>3<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>2<\/mn><mo>,<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>evaluate<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>y<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>y<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>given<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>x<\/mi><mo>=<\/mo><mi>i<\/mi><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>x<\/mi><mo>=<\/mo><mi>i<\/mi><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1570762\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1570763\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1570762-solution\">7<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1463486\">If<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>y<\/mi><mo>=<\/mo><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>5<\/mn><mo>,<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>y<\/mi><mo>=<\/mo><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>5<\/mn><mo>,<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>evaluate<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>y<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>y<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>given<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>x<\/mi><mo>=<\/mo><mn>2<\/mn><mo>+<\/mo><mi>i<\/mi><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>x<\/mi><mo>=<\/mo><mn>2<\/mn><mo>+<\/mo><mi>i<\/mi><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1339584\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1339585\" data-type=\"problem\"><span class=\"os-number\">8<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1339586\">If<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>y<\/mi><mo>=<\/mo><mn>2<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo>,<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>y<\/mi><mo>=<\/mo><mn>2<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo>,<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>evaluate<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>y<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>y<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>given<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>x<\/mi><mo>=<\/mo><mn>2<\/mn><mo>\u2212<\/mo><mn>3<\/mn><mi>i<\/mi><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>x<\/mi><mo>=<\/mo><mn>2<\/mn><mo>\u2212<\/mo><mn>3<\/mn><mi>i<\/mi><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1701288\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1701289\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1701288-solution\">9<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1701290\">If<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>y<\/mi><mo>=<\/mo><mfrac>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn>\n<\/mrow>\n<mrow>\n<mn>2<\/mn><mo>\u2212<\/mo><mi>x<\/mi>\n<\/mrow>\n<\/mfrac>\n<mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>y<\/mi><mo>=<\/mo><mfrac>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn>\n<\/mrow>\n<mrow>\n<mn>2<\/mn><mo>\u2212<\/mo><mi>x<\/mi>\n<\/mrow>\n<\/mfrac>\n<mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>evaluate<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>y<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>y<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>given<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>x<\/mi><mo>=<\/mo><mn>5<\/mn><mi>i<\/mi><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>x<\/mi><mo>=<\/mo><mn>5<\/mn><mi>i<\/mi><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1353461\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id3008607\" data-type=\"problem\"><span class=\"os-number\">10<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id3008608\">If<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>y<\/mi><mo>=<\/mo><mfrac>\n<mrow>\n<mn>1<\/mn><mo>+<\/mo><mn>2<\/mn><mi>x<\/mi>\n<\/mrow>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>3<\/mn>\n<\/mrow>\n<\/mfrac>\n<mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>y<\/mi><mo>=<\/mo><mfrac>\n<mrow>\n<mn>1<\/mn><mo>+<\/mo><mn>2<\/mn><mi>x<\/mi>\n<\/mrow>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>3<\/mn>\n<\/mrow>\n<\/mfrac>\n<mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>evaluate<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>y<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>y<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>given<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>x<\/mi><mo>=<\/mo><mn>4<\/mn><mi>i<\/mi><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>x<\/mi><mo>=<\/mo><mn>4<\/mn><mi>i<\/mi><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1512803\" data-depth=\"2\">\n<h3 data-type=\"title\">Graphical<\/h3>\n<p id=\"fs-id1951802\">For the following exercises, plot the complex numbers on the complex plane.<\/p>\n<div id=\"fs-id2771166\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2771167\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2771166-solution\">11<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2771168\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>1<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>i<\/mi><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>1<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>i<\/mi><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1312026\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1312027\" data-type=\"problem\"><span class=\"os-number\">12<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2048575\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>\u22122<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>\u22122<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1045430\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1403322\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1045430-solution\">13<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1403323\">\n<p><math display=\"inline\"><semantics><mrow>\n<mi>i<\/mi>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mi>i<\/mi><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2513619\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2513620\" data-type=\"problem\"><span class=\"os-number\">14<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2513621\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>\u22123<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>\u22123<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id2644609\" data-depth=\"2\">\n<h3 data-type=\"title\">Numeric<\/h3>\n<p id=\"fs-id2777086\">For the following exercises, perform the indicated operation and express the result as a simplified complex number.<\/p>\n<div id=\"fs-id1296020\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1296021\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1296020-solution\">15<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1296022\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>3<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi><\/mrow>\n<mo>)<\/mo><\/mrow><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mn>5<\/mn><mo>\u2212<\/mo><mn>3<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>3<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi><\/mrow>\n<mo>)<\/mo><\/mrow><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mn>5<\/mn><mo>\u2212<\/mo><mn>3<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1826038\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1826039\" data-type=\"problem\"><span class=\"os-number\">16<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1826040\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>\u22122<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi><\/mrow>\n<mo>)<\/mo><\/mrow><mo>+<\/mo><mrow><mo>(<\/mo>\n<mrow>\n<mn>1<\/mn><mo>+<\/mo><mn>6<\/mn><mi>i<\/mi><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>\u22122<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi><\/mrow>\n<mo>)<\/mo><\/mrow><mo>+<\/mo><mrow><mo>(<\/mo>\n<mrow>\n<mn>1<\/mn><mo>+<\/mo><mn>6<\/mn><mi>i<\/mi><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2441443\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2441444\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2441443-solution\">17<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2778017\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>\u22125<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi><\/mrow>\n<mo>)<\/mo><\/mrow><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>6<\/mn><mo>\u2212<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>\u22125<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi><\/mrow>\n<mo>)<\/mo><\/mrow><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>6<\/mn><mo>\u2212<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2439091\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2496696\" data-type=\"problem\"><span class=\"os-number\">18<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2496697\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>2<\/mn><mo>\u2212<\/mo><mn>3<\/mn><mi>i<\/mi><\/mrow>\n<mo>)<\/mo><\/mrow><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>2<\/mn><mo>\u2212<\/mo><mn>3<\/mn><mi>i<\/mi><\/mrow>\n<mo>)<\/mo><\/mrow><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1976046\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1976047\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1976046-solution\">19<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1976048\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>\u22124<\/mn><mo>+<\/mo><mn>4<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u22126<\/mn><mo>+<\/mo><mn>9<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mo stretchy=\"false\">(<\/mo><mn>\u22124<\/mn><mo>+<\/mo><mn>4<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u22126<\/mn><mo>+<\/mo><mn>9<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1509888\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1509889\" data-type=\"problem\"><span class=\"os-number\">20<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1509890\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>2<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>2<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1562525\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1562526\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1562525-solution\">21<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id3176342\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>5<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>i<\/mi><\/mrow>\n<mo>)<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>5<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>i<\/mi><\/mrow>\n<mo>)<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2505958\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2267959\" data-type=\"problem\"><span class=\"os-number\">22<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2267960\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>6<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>i<\/mi><\/mrow>\n<mo>)<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><mn>5<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>6<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>i<\/mi><\/mrow>\n<mo>)<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><mn>5<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1710164\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1710165\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1710164-solution\">23<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1710166\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>\u22122<\/mn><mo>+<\/mo><mn>4<\/mn><mi>i<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\n<mn>8<\/mn>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>\u22122<\/mn><mo>+<\/mo><mn>4<\/mn><mi>i<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\n<mn>8<\/mn>\n<mo>)<\/mo><\/mrow>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2905259\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1907594\" data-type=\"problem\"><span class=\"os-number\">24<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1907595\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>2<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo>\u2212<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>2<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo>\u2212<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2905821\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2781269\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2905821-solution\">25<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2781270\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>\u22121<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi><\/mrow>\n<mo>)<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><mn>\u22122<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>\u22121<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi><\/mrow>\n<mo>)<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><mn>\u22122<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1832281\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1832282\" data-type=\"problem\"><span class=\"os-number\">26<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1832283\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>4<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>i<\/mi><\/mrow>\n<mo>)<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>4<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>i<\/mi><\/mrow>\n<mo>)<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1894961\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1894962\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1894961-solution\">27<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1894963\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>3<\/mn><mo>+<\/mo><mn>4<\/mn><mi>i<\/mi><\/mrow>\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\n<mrow>\n<mn>3<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>3<\/mn><mo>+<\/mo><mn>4<\/mn><mi>i<\/mi><\/mrow>\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\n<mrow>\n<mn>3<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1919888\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1919889\" data-type=\"problem\"><span class=\"os-number\">28<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1919890\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mrow>\n<mn>3<\/mn><mo>+<\/mo><mn>4<\/mn><mi>i<\/mi>\n<\/mrow>\n<mn>2<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mrow>\n<mn>3<\/mn><mo>+<\/mo><mn>4<\/mn><mi>i<\/mi>\n<\/mrow>\n<mn>2<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1403158\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2293278\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1403158-solution\">29<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2293280\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mrow>\n<mn>6<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>i<\/mi>\n<\/mrow>\n<mn>3<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mrow>\n<mn>6<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>i<\/mi>\n<\/mrow>\n<mn>3<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1576927\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1576928\" data-type=\"problem\"><span class=\"os-number\">30<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1726713\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mrow>\n<mo>\u2212<\/mo><mn>5<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi>\n<\/mrow>\n<mrow>\n<mn>2<\/mn><mi>i<\/mi>\n<\/mrow>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mrow>\n<mo>\u2212<\/mo><mn>5<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi>\n<\/mrow>\n<mrow>\n<mn>2<\/mn><mi>i<\/mi>\n<\/mrow>\n<\/mfrac>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1403309\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1403310\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1403309-solution\">31<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1403311\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mrow>\n<mn>6<\/mn><mo>+<\/mo><mn>4<\/mn><mi>i<\/mi>\n<\/mrow>\n<mi>i<\/mi>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mrow>\n<mn>6<\/mn><mo>+<\/mo><mn>4<\/mn><mi>i<\/mi>\n<\/mrow>\n<mi>i<\/mi>\n<\/mfrac>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1591150\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1228965\" data-type=\"problem\"><span class=\"os-number\">32<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1228966\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mrow>\n<mn>2<\/mn><mo>\u2212<\/mo><mn>3<\/mn><mi>i<\/mi>\n<\/mrow>\n<mrow>\n<mn>4<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi>\n<\/mrow>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mrow>\n<mn>2<\/mn><mo>\u2212<\/mo><mn>3<\/mn><mi>i<\/mi>\n<\/mrow>\n<mrow>\n<mn>4<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi>\n<\/mrow>\n<\/mfrac>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2437570\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1205233\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2437570-solution\">33<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1205234\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mrow>\n<mn>3<\/mn><mo>+<\/mo><mn>4<\/mn><mi>i<\/mi>\n<\/mrow>\n<mrow>\n<mn>2<\/mn><mo>\u2212<\/mo><mi>i<\/mi>\n<\/mrow>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mrow>\n<mn>3<\/mn><mo>+<\/mo><mn>4<\/mn><mi>i<\/mi>\n<\/mrow>\n<mrow>\n<mn>2<\/mn><mo>\u2212<\/mo><mi>i<\/mi>\n<\/mrow>\n<\/mfrac>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2484840\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2691265\" data-type=\"problem\"><span class=\"os-number\">34<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2691266\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mrow>\n<mn>2<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi>\n<\/mrow>\n<mrow>\n<mn>2<\/mn><mo>\u2212<\/mo><mn>3<\/mn><mi>i<\/mi>\n<\/mrow>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mrow>\n<mn>2<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi>\n<\/mrow>\n<mrow>\n<mn>2<\/mn><mo>\u2212<\/mo><mn>3<\/mn><mi>i<\/mi>\n<\/mrow>\n<\/mfrac>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2506948\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1482316\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2506948-solution\">35<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1482317\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msqrt>\n<mrow>\n<mn>\u22129<\/mn>\n<\/mrow>\n<\/msqrt>\n<mo>+<\/mo><mn>3<\/mn><msqrt>\n<mrow>\n<mn>\u221216<\/mn>\n<\/mrow>\n<\/msqrt>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msqrt>\n<mrow>\n<mn>\u22129<\/mn>\n<\/mrow>\n<\/msqrt>\n<mo>+<\/mo><mn>3<\/mn><msqrt>\n<mrow>\n<mn>\u221216<\/mn>\n<\/mrow>\n<\/msqrt>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2048454\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id3008542\" data-type=\"problem\"><span class=\"os-number\">36<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id3008543\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mo>\u2212<\/mo><msqrt>\n<mrow>\n<mn>\u22124<\/mn><\/mrow>\n<\/msqrt>\n<mo>\u2212<\/mo><mn>4<\/mn><msqrt>\n<mrow>\n<mn>\u221225<\/mn><\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mo>\u2212<\/mo><msqrt>\n<mrow>\n<mn>\u22124<\/mn><\/mrow>\n<\/msqrt>\n<mo>\u2212<\/mo><mn>4<\/mn><msqrt>\n<mrow>\n<mn>\u221225<\/mn><\/mrow>\n<\/msqrt>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1823209\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1823210\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1823209-solution\">37<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1560121\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mrow>\n<mn>2<\/mn><mo>+<\/mo><msqrt>\n<mrow>\n<mo>\u2212<\/mo><mn>12<\/mn>\n<\/mrow>\n<\/msqrt>&nbsp;\n<\/mrow>\n<mn>2<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mrow>\n<mn>2<\/mn><mo>+<\/mo><msqrt>\n<mrow>\n<mo>\u2212<\/mo><mn>12<\/mn>\n<\/mrow>\n<\/msqrt>&nbsp;\n<\/mrow>\n<mn>2<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1823150\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2644284\" data-type=\"problem\"><span class=\"os-number\">38<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2644285\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mrow>\n<mn>4<\/mn><mo>+<\/mo><msqrt>\n<mrow>\n<mo>\u2212<\/mo><mn>20<\/mn>\n<\/mrow>\n<\/msqrt>&nbsp;\n<\/mrow>\n<mn>2<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mrow>\n<mn>4<\/mn><mo>+<\/mo><msqrt>\n<mrow>\n<mo>\u2212<\/mo><mn>20<\/mn>\n<\/mrow>\n<\/msqrt>&nbsp;\n<\/mrow>\n<mn>2<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1690727\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1690728\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1690727-solution\">39<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1690729\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>i<\/mi>\n<mn>8<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>i<\/mi>\n<mn>8<\/mn>\n<\/msup>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2995714\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2995715\" data-type=\"problem\"><span class=\"os-number\">40<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id3133451\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>i<\/mi>\n<mrow>\n<mn>15<\/mn>\n<\/mrow>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>i<\/mi>\n<mrow>\n<mn>15<\/mn>\n<\/mrow>\n<\/msup>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1479782\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2512584\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1479782-solution\">41<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2512585\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>i<\/mi>\n<mrow>\n<mn>22<\/mn>\n<\/mrow>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>i<\/mi>\n<mrow>\n<mn>22<\/mn>\n<\/mrow>\n<\/msup>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1537530\" data-depth=\"2\">\n<h3 data-type=\"title\">Technology<\/h3>\n<p id=\"fs-id1353170\">For the following exercises, use a calculator to help answer the questions.<\/p>\n<div id=\"fs-id2980662\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2980663\" data-type=\"problem\"><span class=\"os-number\">42<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2980664\">Evaluate<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>+<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo>\n<\/mrow>\n<mi>k<\/mi>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>+<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo>\n<\/mrow>\n<mi>k<\/mi>\n<\/msup>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math>for<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>k<\/mi><mo>=<\/mo><mn>4<\/mn><mo>,<\/mo><mn>8<\/mn><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>k<\/mi><mo>=<\/mo><mn>4<\/mn><mo>,<\/mo><mn>8<\/mn><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>and<\/p>\n<p><math display=\"inline\"><semantics><mrow><mn>12<\/mn><\/mrow><annotation-xml encoding=\"MathML-Content\"><mn>12<\/mn><\/annotation-xml><\/semantics><\/math>. Predict the value if<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>k<\/mi><mo>=<\/mo><mn>16.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>k<\/mi><mo>=<\/mo><mn>16.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2279285\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2279286\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2279285-solution\">43<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2279287\">Evaluate<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo>\n<\/mrow>\n<mi>k<\/mi>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo>\n<\/mrow>\n<mi>k<\/mi>\n<\/msup>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math>for<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>k<\/mi><mo>=<\/mo><mn>2<\/mn><mo>,<\/mo><mn>6<\/mn><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>k<\/mi><mo>=<\/mo><mn>2<\/mn><mo>,<\/mo><mn>6<\/mn><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>and<\/p>\n<p><math display=\"inline\"><semantics><mrow><mn>10<\/mn><\/mrow><annotation-xml encoding=\"MathML-Content\"><mn>10<\/mn><\/annotation-xml><\/semantics><\/math>. Predict the value if<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>k<\/mi><mo>=<\/mo><mn>14.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>k<\/mi><mo>=<\/mo><mn>14.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1007677\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1007678\" data-type=\"problem\"><span class=\"os-number\">44<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1520536\">Evaluate<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mtext>l<\/mtext><mo>+<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\n<mi>k<\/mi>\n<\/msup>\n<mo>\u2212<\/mo><msup>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mtext>l<\/mtext><mo>\u2212<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\n<mi>k<\/mi>\n<\/msup>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mtext>l<\/mtext><mo>+<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\n<mi>k<\/mi>\n<\/msup>\n<mo>\u2212<\/mo><msup>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mtext>l<\/mtext><mo>\u2212<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\n<mi>k<\/mi>\n<\/msup>\n<\/mrow><\/annotation-xml><\/semantics><\/math>for<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>k<\/mi><mo>=<\/mo><mn>4<\/mn><mo>,<\/mo><mn>8<\/mn><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>k<\/mi><mo>=<\/mo><mn>4<\/mn><mo>,<\/mo><mn>8<\/mn><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>and<\/p>\n<p><math display=\"inline\"><semantics><mrow><mn>12<\/mn><\/mrow><annotation-xml encoding=\"MathML-Content\"><mn>12<\/mn><\/annotation-xml><\/semantics><\/math>. Predict the value for<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>k<\/mi><mo>=<\/mo><mn>16.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>k<\/mi><mo>=<\/mo><mn>16.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2410330\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2410331\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2410330-solution\">45<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2410332\">Show that a solution of<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>6<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>1<\/mn><mo>=<\/mo><mn>0<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>x<\/mi>\n<mn>6<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>1<\/mn><mo>=<\/mo><mn>0<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>is<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mrow>\n<msqrt>\n<mn>3<\/mn>\n<\/msqrt>&nbsp;\n<\/mrow>\n<mn>2<\/mn>\n<\/mfrac>\n<mo>+<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<mi>i<\/mi><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mrow>\n<msqrt>\n<mn>3<\/mn>\n<\/msqrt>&nbsp;\n<\/mrow>\n<mn>2<\/mn>\n<\/mfrac>\n<mo>+<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<mi>i<\/mi><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1385742\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1385743\" data-type=\"problem\"><span class=\"os-number\">46<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1385744\">Show that a solution of<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>8<\/mn>\n<\/msup>\n<mn>\u22121<\/mn><mo>=<\/mo><mn>0<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>x<\/mi>\n<mn>8<\/mn>\n<\/msup>\n<mn>\u22121<\/mn><mo>=<\/mo><mn>0<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>is<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mrow>\n<msqrt>\n<mn>2<\/mn>\n<\/msqrt>&nbsp;\n<\/mrow>\n<mn>2<\/mn>\n<\/mfrac>\n<mo>+<\/mo><mfrac>\n<mrow>\n<msqrt>\n<mn>2<\/mn>\n<\/msqrt>&nbsp;\n<\/mrow>\n<mn>2<\/mn>\n<\/mfrac>\n<mi>i<\/mi><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mrow>\n<msqrt>\n<mn>2<\/mn>\n<\/msqrt>&nbsp;\n<\/mrow>\n<mn>2<\/mn>\n<\/mfrac>\n<mo>+<\/mo><mfrac>\n<mrow>\n<msqrt>\n<mn>2<\/mn>\n<\/msqrt>&nbsp;\n<\/mrow>\n<mn>2<\/mn>\n<\/mfrac>\n<mi>i<\/mi><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id2494010\" data-depth=\"2\">\n<h3 data-type=\"title\">Extensions<\/h3>\n<p id=\"fs-id2371706\">For the following exercises, evaluate the expressions, writing the result as a simplified complex number.<\/p>\n<div id=\"fs-id3161030\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id3161031\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id3161030-solution\">47<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2434972\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mi>i<\/mi>\n<\/mfrac>\n<mo>+<\/mo><mfrac>\n<mn>4<\/mn>\n<mrow>\n<msup>\n<mi>i<\/mi>\n<mn>3<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mn>1<\/mn>\n<mi>i<\/mi>\n<\/mfrac>\n<mo>+<\/mo><mfrac>\n<mn>4<\/mn>\n<mrow>\n<msup>\n<mi>i<\/mi>\n<mn>3<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mfrac>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2794969\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2794970\" data-type=\"problem\"><span class=\"os-number\">48<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2794971\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mrow>\n<msup>\n<mi>i<\/mi>\n<mrow>\n<mn>11<\/mn>\n<\/mrow>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mfrac>\n<mo>\u2212<\/mo><mfrac>\n<mn>1<\/mn>\n<mrow>\n<msup>\n<mi>i<\/mi>\n<mrow>\n<mn>21<\/mn>\n<\/mrow>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mn>1<\/mn>\n<mrow>\n<msup>\n<mi>i<\/mi>\n<mrow>\n<mn>11<\/mn>\n<\/mrow>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mfrac>\n<mo>\u2212<\/mo><mfrac>\n<mn>1<\/mn>\n<mrow>\n<msup>\n<mi>i<\/mi>\n<mrow>\n<mn>21<\/mn>\n<\/mrow>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mfrac>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2716062\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2716064\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2716062-solution\">49<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2515854\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>i<\/mi>\n<mn>7<\/mn>\n<\/msup>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>1<\/mn><mo>+<\/mo><msup>\n<mi>i<\/mi>\n<mn>2<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>i<\/mi>\n<mn>7<\/mn>\n<\/msup>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>1<\/mn><mo>+<\/mo><msup>\n<mi>i<\/mi>\n<mn>2<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2370481\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2943044\" data-type=\"problem\"><span class=\"os-number\">50<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2943045\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>i<\/mi>\n<mrow>\n<mn>\u22123<\/mn>\n<\/mrow>\n<\/msup>\n<mo>+<\/mo><mn>5<\/mn><msup>\n<mi>i<\/mi>\n<mn>7<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>i<\/mi>\n<mrow>\n<mn>\u22123<\/mn>\n<\/mrow>\n<\/msup>\n<mo>+<\/mo><mn>5<\/mn><msup>\n<mi>i<\/mi>\n<mn>7<\/mn>\n<\/msup>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1390725\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1390726\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1390725-solution\">51<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1390727\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>2<\/mn><mo>+<\/mo><mi>i<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\n<mrow>\n<mn>4<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>i<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>+<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo>\n<\/mrow>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>2<\/mn><mo>+<\/mo><mi>i<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\n<mrow>\n<mn>4<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>i<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>+<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo>\n<\/mrow>\n<\/mfrac>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1914286\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1914287\" data-type=\"problem\"><span class=\"os-number\">52<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1945289\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>1<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi><\/mrow>\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\n<mrow>\n<mn>2<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mfrac>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>1<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi><\/mrow>\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\n<mrow>\n<mn>2<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mfrac>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2357467\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2357468\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2357467-solution\">53<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2357469\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mrow>\n<msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>3<\/mn><mo>+<\/mo><mi>i<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<mn>2<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<mrow>\n<msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>1<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<mn>2<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mrow>\n<msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>3<\/mn><mo>+<\/mo><mi>i<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<mn>2<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<mrow>\n<msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>1<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<mn>2<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mfrac>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2294729\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2721846\" data-type=\"problem\"><span class=\"os-number\">54<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2721848\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mrow>\n<mn>3<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi>\n<\/mrow>\n<mrow>\n<mn>2<\/mn><mo>+<\/mo><mi>i<\/mi>\n<\/mrow>\n<\/mfrac>\n<mo>+<\/mo><mrow><mo>(<\/mo>\n<mrow>\n<mn>4<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mrow>\n<mn>3<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi>\n<\/mrow>\n<mrow>\n<mn>2<\/mn><mo>+<\/mo><mi>i<\/mi>\n<\/mrow>\n<\/mfrac>\n<mo>+<\/mo><mrow><mo>(<\/mo>\n<mrow>\n<mn>4<\/mn><mo>+<\/mo><mn>3<\/mn><mi>i<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1758544\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1758545\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1758544-solution\">55<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2932432\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mrow>\n<mn>4<\/mn><mo>+<\/mo><mi>i<\/mi>\n<\/mrow>\n<mi>i<\/mi>\n<\/mfrac>\n<mo>+<\/mo><mfrac>\n<mrow>\n<mn>3<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi>\n<\/mrow>\n<mrow>\n<mn>1<\/mn><mo>\u2212<\/mo><mi>i<\/mi>\n<\/mrow>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mrow>\n<mn>4<\/mn><mo>+<\/mo><mi>i<\/mi>\n<\/mrow>\n<mi>i<\/mi>\n<\/mfrac>\n<mo>+<\/mo><mfrac>\n<mrow>\n<mn>3<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>i<\/mi>\n<\/mrow>\n<mrow>\n<mn>1<\/mn><mo>\u2212<\/mo><mi>i<\/mi>\n<\/mrow>\n<\/mfrac>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id3034376\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id3034377\" data-type=\"problem\"><span class=\"os-number\">56<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2402604\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mrow>\n<mn>3<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi>\n<\/mrow>\n<mrow>\n<mn>1<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi>\n<\/mrow>\n<\/mfrac>\n<mo>\u2212<\/mo><mfrac>\n<mrow>\n<mn>2<\/mn><mo>\u2212<\/mo><mn>3<\/mn><mi>i<\/mi>\n<\/mrow>\n<mrow>\n<mn>3<\/mn><mo>+<\/mo><mi>i<\/mi>\n<\/mrow>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mrow>\n<mn>3<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi>\n<\/mrow>\n<mrow>\n<mn>1<\/mn><mo>+<\/mo><mn>2<\/mn><mi>i<\/mi>\n<\/mrow>\n<\/mfrac>\n<mo>\u2212<\/mo><mfrac>\n<mrow>\n<mn>2<\/mn><mo>\u2212<\/mo><mn>3<\/mn><mi>i<\/mi>\n<\/mrow>\n<mrow>\n<mn>3<\/mn><mo>+<\/mo><mi>i<\/mi>\n<\/mrow>\n<\/mfrac>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/section>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"author":158,"menu_order":4,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-110","chapter","type-chapter","status-publish","hentry"],"part":51,"_links":{"self":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/110","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/users\/158"}],"version-history":[{"count":2,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/110\/revisions"}],"predecessor-version":[{"id":321,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/110\/revisions\/321"}],"part":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/parts\/51"}],"metadata":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/110\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/media?parent=110"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=110"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/contributor?post=110"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/license?post=110"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}