{"id":86,"date":"2025-04-09T17:05:21","date_gmt":"2025-04-09T17:05:21","guid":{"rendered":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/chapter\/1-3-radicals-and-rational-exponents-college-algebra-2e-openstax\/"},"modified":"2025-09-22T19:31:06","modified_gmt":"2025-09-22T19:31:06","slug":"1-3-radicals-and-rational-exponents","status":"publish","type":"chapter","link":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/chapter\/1-3-radicals-and-rational-exponents\/","title":{"raw":"1.3 Radicals and Rational Exponents","rendered":"1.3 Radicals and Rational Exponents"},"content":{"raw":"<div id=\"main-content\" class=\"MainContent__ContentStyles-sc-6yy1if-0 NnXKu\" tabindex=\"-1\" data-dynamic-style=\"true\">\r\n<div id=\"page_1834bc64-7094-4df1-a530-b3166a295697\" class=\"chapter-content-module\" data-type=\"page\" data-book-content=\"true\">\r\n<div class=\"textbox textbox--learning-objectives\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Learning Objectives<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nIn this section, you will:\r\n<ul>\r\n \t<li>Evaluate square roots.<\/li>\r\n \t<li>Use the product rule to simplify square roots.<\/li>\r\n \t<li>Use the quotient rule to simplify square roots.<\/li>\r\n \t<li>Add and subtract square roots.<\/li>\r\n \t<li>Rationalize denominators.<\/li>\r\n \t<li>Use rational roots.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167339431492\">A hardware store sells 16-ft ladders and 24-ft ladders. A window is located 12 feet above the ground. A ladder needs to be purchased that will reach the window from a point on the ground 5 feet from the building. To find out the length of ladder needed, we can draw a right triangle as shown in Figure 1, and use the Pythagorean Theorem.<\/p>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_1430\" align=\"aligncenter\" width=\"219\"]<img class=\"size-full wp-image-1430\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.3-fig-1.webp\" alt=\"\" width=\"219\" height=\"297\" \/> Figure 1[\/caption]\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rcl}a^2 + b^2 &amp;=&amp; c^2 \\\\5^2 + 12^2 &amp;=&amp; c^2 \\\\169 &amp;=&amp; c^2\\end{array} [\/latex]<\/p>\r\n\r\n<div id=\"page_1834bc64-7094-4df1-a530-b3166a295697\" class=\"chapter-content-module\" data-type=\"page\" data-book-content=\"true\">\r\n<p id=\"fs-id1167339431629\">Now, we need to find out the length that, when squared, is 169, to determine which ladder to choose. In other words, we need to find a square root. In this section, we will investigate methods of finding solutions to problems such as this one.<\/p>\r\n\r\n<section id=\"fs-id1167339431634\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Evaluating Square Roots<\/h2>\r\n<p id=\"fs-id1167339431640\">When the square root of a number is squared, the result is the original number. Since [latex] 4^2=16, [\/latex] the square root of [latex] 16 [\/latex] is [latex] 4. [\/latex] The square root function is the inverse of the squaring function just as subtraction is the inverse of addition. To undo squaring, we take the square root.<\/p>\r\n<p id=\"fs-id1484606\">In general terms, if [latex] a [\/latex] is a positive real number, then the square root of [latex] a [\/latex] is a number that, when multiplied by itself, gives [latex] a. [\/latex] The square root could be positive or negative because multiplying two negative numbers gives a positive number. The <strong><span id=\"term-00006\" data-type=\"term\">principal square root<\/span><\/strong> is the nonnegative number that when multiplied by itself equals [latex] a. [\/latex] The square root obtained using a calculator is the principal square root.<\/p>\r\n<p id=\"fs-id1507362\">The principal square root of [latex] a [\/latex] is written as [latex] \\sqrt{a}. [\/latex] The symbol is called a <strong><span id=\"term-00007\" data-type=\"term\">radical<\/span><\/strong>, the term under the symbol is called the <strong><span id=\"term-00008\" data-type=\"term\">radicand<\/span><\/strong>, and the entire expression is called a <strong><span id=\"term-00009\" data-type=\"term\">radical expression<\/span><\/strong>.<span id=\"fs-id1495127\" data-type=\"media\" data-alt=\"The expression: square root of twenty-five is enclosed in a circle. The circle has an arrow pointing to it labeled: Radical expression. The square root symbol has an arrow pointing to it labeled: Radical. The number twenty-five has an arrow pointing to it labeled: Radicand.\"><\/span><\/p>\r\n<img class=\"size-medium wp-image-1431 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.3-rad-300x129.webp\" alt=\"\" width=\"300\" height=\"129\" \/>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Principal Square Root<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nThe <span id=\"term-00010\" data-type=\"term\">principal square root<\/span> of\u00a0[latex] a [\/latex] is the nonnegative number that, when multiplied by itself, equals\u00a0[latex] a. [\/latex] It is written as a <span id=\"term-00011\" data-type=\"term\">radical expression<\/span><strong>,<\/strong> with a symbol called a <span id=\"term-00012\" data-type=\"term\">radical<\/span> over the term called the <span id=\"term-00013\" data-type=\"term\">radicand<\/span>: [latex] \\sqrt{a}. [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Q&amp;A<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Q: Does<\/strong> [latex] \\sqrt{25}=\\pm5? [\/latex]\r\n\r\n<em>A: No. Although both\u00a0[latex] 5^2 [\/latex] and\u00a0[latex] (-5)^2 [\/latex] are\u00a0[latex] 25, [\/latex] the radical symbol implies only a nonnegative root, the principal square root. The principal square root of 25 is<\/em> [latex] \\sqrt{25}=5. [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 1: Evaluating Square Roots<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nEvaluate each expression.\r\n\r\n(a) [latex] \\sqrt{100} [\/latex]\r\n\r\n(b) [latex] \\sqrt{\\sqrt{16}} [\/latex]\r\n\r\n(c) [latex] \\sqrt{25+144} [\/latex]\r\n\r\n(d) [latex] \\sqrt{49}-\\sqrt{81} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>(a) [latex] \\sqrt{100}=10 [\/latex] because [latex] 10^2=100 [\/latex]\r\n\r\n&nbsp;\r\n\r\n(b) [latex] \\sqrt{\\sqrt{16}}=\\sqrt{4} [\/latex] because [latex] 4^2=16 [\/latex] and [latex] 2^2=4 [\/latex]\r\n\r\n&nbsp;\r\n\r\n(c) [latex] \\sqrt{25+144}=\\sqrt{169}=13 [\/latex] because [latex] 13^2=169 [\/latex]\r\n\r\n&nbsp;\r\n\r\n(d) [latex] \\sqrt{49}-\\sqrt{81}=7-9=-2 [\/latex] because [latex] 7^2=49 [\/latex] and [latex] 9^2=81 [\/latex]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Q&amp;A<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Q: For\u00a0[latex] \\sqrt{25+144}, [\/latex] can we find the square roots before adding?<\/strong>\r\n\r\n<em>A: No.\u00a0[latex] \\sqrt{25}+\\sqrt{144}=5+12=17. [\/latex] This is not equivalent to\u00a0[latex] \\sqrt{25+144}=13. [\/latex] The order of operations requires us to add the terms in the radicand before finding the square root.<\/em>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nEvaluate each expression.\r\n\r\n(a) [latex] \\sqrt{225} [\/latex]\r\n\r\n(b) [latex] \\sqrt{\\sqrt{81}} [\/latex]\r\n\r\n(c) [latex] \\sqrt{25-9} [\/latex]\r\n\r\n(d) [latex] \\sqrt{36}+\\sqrt{121} [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<section id=\"fs-id1424126\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Using the Product Rule to Simplify Square Roots<\/h2>\r\n<p id=\"fs-id1499813\">To simplify a square root, we rewrite it such that there are no perfect squares in the radicand. There are several properties of square roots that allow us to simplify complicated radical expressions. The first rule we will look at is the <em data-effect=\"italics\">product rule for simplifying square roots,<\/em> which allows us to separate the square root of a product of two numbers into the product of two separate rational expressions. For instance, we can rewrite [latex] \\sqrt{15} [\/latex] as [latex] \\sqrt{3}\\cdot \\sqrt{5} [\/latex] We can also use the product rule to express the product of multiple radical expressions as a single radical expression.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">The Product Rule for Simplifying Square Roots<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<p id=\"eip-id1167336040491\">If [latex] a [\/latex] and [latex] b [\/latex] are nonnegative, the square root of the product [latex] ab [\/latex] is equal to the product of the square roots of [latex] a [\/latex] and [latex] b. [\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex] \\sqrt{ab}=\\sqrt{a}\\cdot \\sqrt{b} [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">How To<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<p id=\"fs-id1343706\"><strong>Given a square root radical expression, use the product rule to simplify it.\r\n<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1516957\" type=\"1\">\r\n \t<li>Factor any perfect squares from the radicand.<\/li>\r\n \t<li>Write the radical expression as a product of radical expressions.<\/li>\r\n \t<li>Simplify.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 2: Using the Product Rule to Simplify Square Roots<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSimplify the radical expression.\r\n\r\n(a) [latex] \\sqrt{300} [\/latex]\r\n\r\n(b) [latex] \\sqrt{162a^5b^4} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>(a)\r\n\r\n[latex] \\begin{array}{ll}\\sqrt{100 \\cdot 3} &amp; \\quad \\text{Factor perfect square from radicand.} \\\\\\sqrt{100} \\cdot \\sqrt{3} &amp; \\quad \\text{Write radical expression as product of radical expressions.} \\\\10\\sqrt{3} &amp; \\quad \\text{Simplify.}\\end{array} [\/latex]\r\n\r\n&nbsp;\r\n\r\n(b)\r\n\r\n[latex] \\begin{array}{ll}\\sqrt{81a^4b^4 \\cdot 2a} &amp; \\quad \\text{Factor perfect square from radicand.} \\\\\\sqrt{81a^4b^4} \\cdot \\sqrt{2a} &amp; \\quad \\text{Write radical expression as product of radical expressions.} \\\\9a^2b^2\\sqrt{2a} &amp; \\quad \\text{Simplify.}\\end{array} [\/latex]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/section>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSimplify [latex] \\sqrt{50x^2y^3z}. [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">How To<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<p id=\"fs-id537400\"><strong>Given the product of multiple radical expressions, use the product rule to combine them into one radical expression.\r\n<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1343311\" type=\"1\">\r\n \t<li>Express the product of multiple radical expressions as a single radical expression.<\/li>\r\n \t<li>Simplify.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 3: Using the Product Rule to Simplify the Product of Multiple Square Roots<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSimplify the radical expression.\r\n\r\n[latex] \\sqrt{12}\\cdot \\sqrt{3} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{ll}\\sqrt{12 \\cdot 3} &amp; \\quad \\text{Express the product as a single radical expression.} \\\\\\sqrt{36} &amp; \\quad \\text{Simplify.} \\\\6 &amp; \\end{array} [\/latex]<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #3<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSimplify [latex] \\sqrt{50x}\\cdot \\sqrt{2x} [\/latex] assuming [latex] x&gt; 0. [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<section id=\"fs-id1409429\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Using the Quotient Rule to Simplify Square Roots<\/h2>\r\n<p id=\"fs-id1402495\">Just as we can rewrite the square root of a product as a product of square roots, so too can we rewrite the square root of a quotient as a quotient of square roots, using the <em data-effect=\"italics\">quotient rule for simplifying square roots.<\/em> It can be helpful to separate the numerator and denominator of a fraction under a radical so that we can take their square roots separately. We can rewrite [latex] \\sqrt{\\frac{5}{2}} [\/latex] as [latex] \\frac{\\sqrt{5}}{\\sqrt{2}}. [\/latex]<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">The Quotient Rule for Simplifying Square Roots<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nThe square root of the quotient [latex] \\frac{a}{b} [\/latex] is equal to the quotient of the square roots of [latex] a [\/latex] and [latex] b, [\/latex] where [latex] b\\not= 0. [\/latex]\r\n<p style=\"text-align: center;\">[latex] \\sqrt{\\frac{a}{b}}=\\frac{\\sqrt{a}}{\\sqrt{b}} [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">How To<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<p id=\"fs-id1548985\"><strong>Given a radical expression, use the quotient rule to simplify it.\r\n<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1537621\" type=\"1\">\r\n \t<li>Write the radical expression as the quotient of two radical expressions.<\/li>\r\n \t<li>Simplify the numerator and denominator.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 4: Using the Quotient Rule to Simplify Square Roots<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSimplify the radical expression.\r\n\r\n[latex] \\sqrt{\\frac{5}{36}} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{ll}\\frac{\\sqrt{5}}{\\sqrt{36}} &amp; \\quad \\text{Write as a quotient of two radical expressions.} \\\\ \\frac{\\sqrt{5}}{6} &amp; \\quad \\text{Simplify denominator.} \\end{array} [\/latex]<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/section>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #4<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSimplify [latex] \\sqrt{\\frac{2x^2}{9y^4}} [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 5: Using the Quotient Rule to Simplify an Expression with Two Square Roots<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSimplify the radical expression.\r\n\r\n[latex] \\frac{\\sqrt{234x^{11}y}}{\\sqrt{26x^7y}} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>\r\n<p style=\"text-align: left;\">[latex] \\begin{array}{ll}\\sqrt{\\frac{234x^{11}y}{26x^7y}} &amp; \\quad \\text{Combine numerator and denominator into one radical expression.} \\\\ \\sqrt{9x^4} &amp; \\quad \\text{Simplify fraction.} \\\\ 3x^2 &amp; \\quad \\text{Simplify square root.} \\end{array} [\/latex]<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #5<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSimplify [latex]\u00a0 \\frac{\\sqrt{9a^5b^{14}}}{\\sqrt{3a^4b^5}}. [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<section id=\"fs-id1413677\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Adding and Subtracting Square Roots<\/h2>\r\n<p id=\"fs-id1483382\">We can add or subtract radical expressions only when they have the same radicand and when they have the same radical type such as square roots. For example, the sum of [latex] \\sqrt{2} [\/latex] and [latex] 3\\sqrt{2} [\/latex] is [latex] 4\\sqrt{2}. [\/latex] However, it is often possible to simplify radical expressions, and that may change the radicand. The radical expression [latex] \\sqrt{18} [\/latex] can be written with a [latex] 2 [\/latex] in the radicand, as [latex] 3\\sqrt{2}, [\/latex] so [latex] \\sqrt{2}+\\sqrt{18}=\\sqrt{2}+3\\sqrt{2}=4\\sqrt{2}. [\/latex]<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">How To<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<p id=\"fs-id1353954\"><strong>Given a radical expression requiring addition or subtraction of square roots, simplify.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1429237\" type=\"1\">\r\n \t<li>Simplify each radical expression.<\/li>\r\n \t<li>Add or subtract expressions with equal radicands.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 6: Adding Square Roots<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nAdd [latex] 5\\sqrt{12}+2\\sqrt{3}. [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>\r\n<p id=\"fs-id1537392\">We can rewrite [latex] 5\\sqrt{12} [\/latex] as [latex] 5\\sqrt{4\\cdot 3}. [\/latex] According the product rule, this becomes [latex] 5\\sqrt{4}\\sqrt{3}. [\/latex] The square root of [latex] \\sqrt{4} [\/latex] is 2, so the expression becomes [latex] 5(2)\\sqrt{3}, [\/latex] which is [latex] 10\\sqrt{3}. [\/latex] Now the terms have the same radicand so we can add.<\/p>\r\n<p style=\"text-align: center;\">[latex] 10\\sqrt{3}+2\\sqrt{3}=12\\sqrt{3} [\/latex]<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/section>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #6<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nAdd [latex] \\sqrt{5}+6\\sqrt{20} [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 7: Subtracting Square Roots<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSubtract [latex] 20\\sqrt{72a^3b^4c}-14\\sqrt{8a^3b^4c}. [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>Factor 9 out of the first term so that both terms have equal radicands.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rclcl}20\\sqrt{72a^3b^4c} &amp;=&amp; 20\\sqrt{9 \\cdot 8a^3b^4c} &amp;=&amp; 20\\sqrt{9}\\sqrt{8a^3b^4c} \\\\&amp;=&amp; 20(3)\\sqrt{8a^3b^4c} &amp;=&amp; 60\\sqrt{8a^3b^4c}\\end{array} [\/latex]<\/p>\r\nSo\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{l}20\\sqrt{72a^3b^4c} - 14\\sqrt{8a^3b^4c} \\\\= 60\\sqrt{8a^3b^4c} - 14\\sqrt{8a^3b^4c} \\\\= 46\\sqrt{8a^3b^4c}\\end{array} [\/latex]<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #7<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSubtract [latex] 3\\sqrt{80x}-4\\sqrt{45x}. [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<section id=\"fs-id1483223\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Rationalizing Denominators<\/h2>\r\n<p id=\"fs-id1483228\">When an expression involving square root radicals is written in simplest form, it will not contain a radical in the denominator. We can remove radicals from the denominators of fractions using a process called <em data-effect=\"italics\">rationalizing the denominator<\/em>.<\/p>\r\n<p id=\"fs-id1517983\">We know that multiplying by 1 does not change the value of an expression. We use this property of multiplication to change expressions that contain radicals in the denominator. To remove radicals from the denominators of fractions, multiply by the form of 1 that will eliminate the radical.<\/p>\r\n<p id=\"fs-id1517989\">For a denominator containing a single term, multiply by the radical in the denominator over itself. In other words, if the denominator is [latex] b\\sqrt{c}, [\/latex] multiply by [latex] \\frac{\\sqrt{c}}{\\sqrt{c}}. [\/latex]<\/p>\r\n<p id=\"fs-id1484146\">For a denominator containing the sum or difference of a rational and an irrational term, multiply the numerator and denominator by the conjugate of the denominator, which is found by changing the sign of the radical portion of the denominator. If the denominator is [latex] a+b\\sqrt{c}, [\/latex] then the conjugate is [latex] a-b\\sqrt{c}. [\/latex]<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">How To<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<p id=\"fs-id1507281\"><strong>Given an expression with a single square root radical term in the denominator, rationalize the denominator.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1507286\" type=\"a\">\r\n \t<li>Multiply the numerator and denominator by the radical in the denominator.<\/li>\r\n \t<li>Simplify.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 8: Rationalizing a Denominator Containing a Single Term<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWrite [latex] \\frac{2\\sqrt{3}}{3\\sqrt{10}} [\/latex] in simplest form.\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>The radical in the denominator is [latex] \\sqrt{10}. [\/latex] So multiply the fraction by [latex] \\frac{\\sqrt{10}}{\\sqrt{10}}. [\/latex] Then simplify.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{l}\\frac{2\\sqrt{3}}{3\\sqrt{10}} \\cdot \\frac{\\sqrt{10}}{\\sqrt{10}} \\quad \\\\\\frac{2\\sqrt{30}}{30} \\quad \\\\\\frac{\\sqrt{30}}{15}\\end{array} [\/latex]<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/section>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #8<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWrite [latex] \\frac{12\\sqrt{3}}{\\sqrt{2}} [\/latex] in simplest form.\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">How To<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<p id=\"fs-id1167339431931\"><strong>Given an expression with a radical term and a constant in the denominator, rationalize the denominator.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1167339431936\" type=\"1\">\r\n \t<li>Find the conjugate of the denominator.<\/li>\r\n \t<li>Multiply the numerator and denominator by the conjugate.<\/li>\r\n \t<li>Use the distributive property.<\/li>\r\n \t<li>Simplify.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 9: Rationalizing a Denominator Containing Two Terms<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWrite [latex] \\frac{4}{1+\\sqrt{5}} [\/latex] in simplest form.\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>Begin by finding the conjugate of the denominator by writing the denominator and changing the sign. So the conjugate of [latex] 1+\\sqrt{5} [\/latex] is [latex] 1-\\sqrt{5}. [\/latex] Then multiply the fraction by [latex] \\frac{1-\\sqrt{5}}{1-\\sqrt{5}}. [\/latex]\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{ll}\\frac{4}{1 + \\sqrt{5}} \\cdot \\frac{1 - \\sqrt{5}}{1 - \\sqrt{5}} &amp; \\\\\\frac{4 - 4\\sqrt{5}}{-4} &amp; \\quad \\text{Use the distributive property.} \\\\\\sqrt{5} - 1 &amp; \\quad \\text{Simplify.}\\end{array} [\/latex]<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #9<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWrite [latex] \\frac{7}{2+\\sqrt{3}} [\/latex] in simplest form.\r\n\r\n<\/div>\r\n<\/div>\r\n<section id=\"fs-id1167339432042\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Using Rational Roots<\/h2>\r\n<p id=\"fs-id1167339432047\">Although square roots are the most common rational roots, we can also find cube roots, 4th roots, 5th roots, and more. Just as the square root function is the inverse of the squaring function, these roots are the inverse of their respective power functions. These functions can be useful when we need to determine the number that, when raised to a certain power, gives a certain number.<\/p>\r\n\r\n<section id=\"fs-id1167339432053\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Understanding <em data-effect=\"italics\">n<\/em>th Roots<\/h3>\r\n<p id=\"fs-id1167339432064\">Suppose we know that [latex] a^3=8. [\/latex] We want to find what number raised to the 3rd power is equal to 8. Since [latex] 2^3=8, [\/latex] we say that 2 is the cube root of 8.<\/p>\r\n<p id=\"fs-id1167339432119\">The <em data-effect=\"italics\">n<\/em>th root of [latex] a [\/latex] is a number that, when raised to the <em data-effect=\"italics\">n<\/em>th power, gives [latex] a. [\/latex] For example, [latex] -3 [\/latex] is the 5th root of [latex] -243 [\/latex] because [latex] (-3)^5=-243. [\/latex] If [latex] a [\/latex] is a real number with at least one <em data-effect=\"italics\">n<\/em>th root, then the <strong>principal <em data-effect=\"italics\">n<\/em>th root<\/strong> of [latex] a [\/latex] is the number with the same sign as [latex] a [\/latex] that, when raised to the <em data-effect=\"italics\">n<\/em>th power, equals [latex] a. [\/latex]<\/p>\r\n<p id=\"fs-id1167339434352\">The principal <em data-effect=\"italics\">n<\/em>th root of [latex] a [\/latex] is written as [latex] \\sqrt[n]{a}, [\/latex] where [latex] n [\/latex] is a positive integer greater than or equal to 2. In the radical expression, [latex] n [\/latex] is called the <strong>index<\/strong> of the radical.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Principal <em>n<\/em>th Root<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<p id=\"fs-id1167339434411\">If [latex] a [\/latex] is a real number with at least one <em data-effect=\"italics\">n<\/em>th root, then the <span id=\"term-00016\" data-type=\"term\"><strong>principal<\/strong> <em data-effect=\"italics\">n<\/em>th root<\/span> of [latex] a, [\/latex] written as [latex] \\sqrt[n]{a}, [\/latex] is the number with the same sign as [latex] a [\/latex] that, when raised to the <em data-effect=\"italics\">n<\/em>th power, equals [latex] a. [\/latex] The <strong>index <\/strong>of the radical is [latex] n. [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/section>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 10: Simplifying <em>n<\/em>th Roots<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSimplify each of the following:\r\n\r\n(a) [latex] \\sqrt[5]{-32} [\/latex]\r\n\r\n(b) [latex] \\sqrt[4]{4}\\cdot \\sqrt[4]{1,024} [\/latex]\r\n\r\n(c) [latex] -\\sqrt[3]{\\frac{8x^6}{125}} [\/latex]\r\n\r\n(d) [latex] 8\\sqrt[4]{3}-\\sqrt[4]{48} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>(a) [latex] \\sqrt[5]{-32}=-2 [\/latex] because [latex] (-2)^5=-32 [\/latex]\r\n\r\n&nbsp;\r\n\r\n(b) First, express the product as a single radical expression. because [latex] \\sqrt[4]{4,096}=8 [\/latex] because [latex] 8^4=4,096 [\/latex]\r\n\r\n&nbsp;\r\n\r\n(c)\r\n\r\n[latex] \\begin{array}{ll}\\frac{-\\sqrt[3]{8x^6}}{\\sqrt[3]{125}} &amp; \\quad \\text{Write as quotient of two radical expressions.} \\\\\\frac{-2x^2}{5} &amp; \\quad \\text{Simplify.}\\end{array} [\/latex]\r\n\r\n&nbsp;\r\n\r\n(d)\r\n\r\n[latex] \\begin{array}{ll}8\\sqrt[4]{3} - 2\\sqrt[4]{3} &amp; \\quad \\text{Simplify to get equal radicands.} \\\\6\\sqrt[4]{3} &amp; \\quad \\text{Add.}\\end{array} [\/latex]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #10<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSimplify.\r\n\r\n(a) [latex] \\sqrt[3]{-216} [\/latex]\r\n\r\n(b) [latex] \\frac{3\\sqrt[4]{80}}{\\sqrt[4]{5}} [\/latex]\r\n\r\n(c) [latex] 6\\sqrt[3]{9,000}+7\\sqrt[3]{576} [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<section id=\"fs-id1538427\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Using Rational Exponents<\/h3>\r\n<p id=\"fs-id1538433\"><strong>Radical expressions <\/strong>can also be written without using the radical symbol. We can use rational (fractional) exponents. The index must be a positive integer. If the index [latex] n [\/latex] is even, then [latex] a [\/latex] cannot be negative.<\/p>\r\n<p style=\"text-align: center;\">[latex] a^{\\frac{1}{n}}=\\sqrt[n]{a} [\/latex]<\/p>\r\n<p id=\"fs-id1508564\">We can also have rational exponents with numerators other than 1. In these cases, the exponent must be a fraction in lowest terms. We raise the base to a power and take an <em data-effect=\"italics\">n<\/em>th root. The numerator tells us the power and the denominator tells us the root.<\/p>\r\n<p style=\"text-align: center;\">[latex] a^{\\frac{m}{n}}=\\left(\\sqrt[n]{a}\\right)^m=\\sqrt[n]{a^m} [\/latex]<\/p>\r\n<p id=\"fs-id1529202\">All of the properties of exponents that we learned for integer exponents also hold for rational exponents.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Rational Exponents<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nRational exponents are another way to express principal <em data-effect=\"italics\">n<\/em>th roots. The general form for converting between a radical expression with a radical symbol and one with a rational exponent is\r\n<p style=\"text-align: center;\">[latex] a^{\\frac{m}{n}}=\\left(\\sqrt[n]{a}\\right)^m=\\sqrt[n]{a^m}\u00a0 [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">How To<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<p id=\"eip-id1592533\"><strong>Given an expression with a rational exponent, write the expression as a radical.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1482550\" type=\"1\">\r\n \t<li>Determine the power by looking at the numerator of the exponent.<\/li>\r\n \t<li>Determine the root by looking at the denominator of the exponent.<\/li>\r\n \t<li>Using the base as the radicand, raise the radicand to the power and use the root as the index.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 11: Writing Rational Exponents as Radicals<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWrite [latex] 343^{\\frac{2}{3}} [\/latex] as a radical. Simplify.\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>The 2 tells us the power and the 3 tells us the root.\r\n<p style=\"text-align: center;\">[latex] 343^{\\frac{2}{3}}=\\left(\\sqrt[3]{343}\\right)^2=\\sqrt[3]{343^2} [\/latex]<\/p>\r\nWe know that [latex] \\sqrt[3]{343}=7 [\/latex] because [latex] 7^3=343. [\/latex] Because the cube root is easy to find, it is easiest to find the cube root before squaring for this problem. In general, it is easier to find the root first and then raise it to a power.\r\n<p style=\"text-align: center;\">[latex] 343^{\\frac{2}{3}}=\\left(\\sqrt[3]{343}\\right)^2=7^2=49 [\/latex]<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/section>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #11<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWrite [latex] 9^{\\frac{5}{2}} [\/latex] as a radical. Simplify.\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 12: Writing Radicals as Rational Exponents<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWrite [latex] \\frac{4}{\\sqrt[7]{a^2}} [\/latex] using a rational exponent.\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>The power is 2 and the root is 7, so the rational exponent will be [latex] \\frac{2}{7}. [\/latex] We get [latex] \\frac{4}{a^{\\frac{2}{7}}}. [\/latex] Using properties of exponents, we get [latex] \\frac{4}{\\sqrt[7]{a^2}}=4a^{\\frac{-2}{7}}. [\/latex]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #12<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWrite [latex] x\\sqrt{(5y)^9} [\/latex] using a rational exponent.\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 13: Simplifying Rational Exponents<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSimplify:\r\n\r\n(a) [latex] 5\\left(2x^{\\frac{3}{4}}\\right)\\left(3x^{\\frac{1}{5}}\\right) [\/latex]\r\n\r\n(b) [latex] \\left(\\frac{16}{9}\\right)^{-\\frac{1}{2}} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>(a)\r\n\r\n[latex] \\begin{array}{ll}30x^{\\frac{3}{4}}x^{\\frac{1}{5}} &amp; \\quad \\text{Multiply the coefficients.} \\\\30x^{\\frac{3}{4} + \\frac{1}{5}} &amp; \\quad \\text{Use properties of exponents.} \\\\30x^{\\frac{19}{20}} &amp; \\quad \\text{Simplify.}\\end{array} [\/latex]\r\n\r\n&nbsp;\r\n\r\n(b)\r\n\r\n[latex] \\begin{array}{ll} \\left(\\frac{9}{16}\\right)^{\\frac{1}{2}} &amp; \\quad \\text{ Use definition of negative exponents.} \\\\\\sqrt{\\frac{9}{16}} &amp; \\quad \\text{ Rewrite as a radical.} \\\\\\frac{\\sqrt{9}}{\\sqrt{16}} &amp; \\quad \\text{ Use the quotient rule.} \\\\\\frac{3}{4} &amp; \\quad \\text{ Simplify.}\\end{array} [\/latex]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #13<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSimplify [latex] (8x)^{\\frac{1}{3}}\\left(14x^{\\frac{6}{5}}\\right). [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Media<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<p id=\"fs-id1487284\">Access these online resources for additional instruction and practice with radicals and rational exponents.<\/p>\r\n\r\n<ul id=\"fs-id1476938\">\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/introradical\" target=\"_blank\" rel=\"noopener nofollow\">Radicals<\/a><\/li>\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/rationexpon\" target=\"_blank\" rel=\"noopener nofollow\">Rational Exponents<\/a><\/li>\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/simpradical\" target=\"_blank\" rel=\"noopener nofollow\">Simplify Radicals<\/a><\/li>\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/rationdenom\" target=\"_blank\" rel=\"noopener nofollow\">Rationalize Denominator<\/a><\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\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\">1.3 Section Exercises<\/span><\/h2>\r\n<section id=\"fs-id1499197\" class=\"section-exercises\" data-depth=\"1\"><section id=\"fs-id1499203\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Verbal<\/h3>\r\n<div id=\"fs-id1499209\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1499210\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1499209-solution\">1<\/a><span class=\"os-divider\">. <\/span>What does it mean when a radical does not have an index? Is the expression equal to the radicand? Explain.\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1538359\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1538360\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">2<\/span><span class=\"os-divider\">. <\/span>Where would radicals come in the order of operations? Explain why.\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1538365\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1538366\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1538365-solution\">3<\/a><span class=\"os-divider\">. <\/span>Every number will have two square roots. What is the principal square root?\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1538374\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1538375\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">4<\/span><span class=\"os-divider\">. <\/span>Can a radical with a negative radicand have a real square root? Why or why not?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1538381\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Numeric<\/h3>\r\n<p id=\"fs-id1538386\">For the following exercises, simplify each expression.<\/p>\r\n\r\n<div id=\"fs-id1514811\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1514812\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1514811-solution\">5<\/a><span class=\"os-divider\">.<\/span> [latex] \\sqrt{256} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1514836\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1514837\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">6<\/span><span class=\"os-divider\">.<\/span> [latex] \\sqrt{\\sqrt{256}} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1499788\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1499789\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1499788-solution\">7<\/a><span class=\"os-divider\">.<\/span> [latex] \\sqrt{4(9+16)} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1549261\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1549262\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">8<\/span><span class=\"os-divider\">.<\/span> [latex] \\sqrt{289}-\\sqrt{121} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1529781\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1529782\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1529781-solution\">9<\/a><span class=\"os-divider\">.<\/span> [latex] \\sqrt{196} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1430084\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1430085\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">10<\/span><span class=\"os-divider\">.<\/span> [latex] \\sqrt{1} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1430101\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1430102\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1430101-solution\">11<\/a><span class=\"os-divider\">.<\/span> [latex] \\sqrt{98} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1487491\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1487492\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">12<\/span><span class=\"os-divider\">.<\/span> [latex] \\sqrt{\\frac{27}{64}} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1545152\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1545153\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1545152-solution\">13<\/a><span class=\"os-divider\">.<\/span> [latex] \\sqrt{\\frac{81}{5}} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1469422\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1469424\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">14<\/span><span class=\"os-divider\">.<\/span> [latex] \\sqrt{800} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1469443\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1469444\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1469443-solution\">15<\/a><span class=\"os-divider\">.<\/span> [latex] \\sqrt{169}+\\sqrt{144} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339435248\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1167339435249\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">16<\/span><span class=\"os-divider\">.<\/span> [latex] \\sqrt{\\frac{8}{50}} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339435278\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1167339435279\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339435278-solution\">17<\/a><span class=\"os-divider\">.<\/span> [latex] \\frac{18}{\\sqrt{162}} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339435349\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1167339435350\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">18<\/span><span class=\"os-divider\">.<\/span> [latex] \\sqrt{192} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339435370\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1167339435371\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339435370-solution\">19<\/a><span class=\"os-divider\">.<\/span> [latex] 14\\sqrt{6}-6\\sqrt{24} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339435421\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1167339435422\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">20<\/span><span class=\"os-divider\">.<\/span> [latex] 15\\sqrt{5}+7\\sqrt{45} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339435454\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1167339435455\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339435454-solution\">21<\/a><span class=\"os-divider\">.<\/span> [latex] \\sqrt{150} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339435492\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1167339435493\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">22<\/span><span class=\"os-divider\">.<\/span> [latex] \\sqrt{\\frac{96}{100}} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339435525\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1167339435526\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339435525-solution\">23<\/a><span class=\"os-divider\">.<\/span> [latex] \\left(\\sqrt{42}\\right) \\left(\\sqrt{30}\\right) [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339435597\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1167339435598\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">24<\/span><span class=\"os-divider\">.<\/span> [latex] 12\\sqrt{3}-4\\sqrt{75} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339435630\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1167339435631\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339435630-solution\">25<\/a><span class=\"os-divider\">.<\/span> [latex] \\sqrt{\\frac{4}{225}} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339435682\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1167339435683\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">26<\/span><span class=\"os-divider\">.<\/span> [latex] \\sqrt{\\frac{405}{324}} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339435715\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1167339435716\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339435715-solution\">27<\/a><span class=\"os-divider\">.<\/span> [latex] \\sqrt{\\frac{360}{361}} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339435782\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1167339435784\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">28<\/span><span class=\"os-divider\">.<\/span> [latex] \\frac{5}{1+\\sqrt{3}} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339435813\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1167339435814\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339435813-solution\">29<\/a><span class=\"os-divider\">.<\/span> [latex] \\frac{8}{1-\\sqrt{17}} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339435883\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1167339435884\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">30<\/span><span class=\"os-divider\">.<\/span> [latex] \\sqrt[4]{16} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339435906\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1167339435907\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339435906-solution\">31<\/a><span class=\"os-divider\">.<\/span> [latex] \\sqrt[3]{128}+3\\sqrt[3]{2} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339435965\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1167339435966\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">32<\/span><span class=\"os-divider\">.<\/span> [latex] \\sqrt[5]{\\frac{-32}{243}} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339436003\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1167339436004\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339436003-solution\">33<\/a><span class=\"os-divider\">.<\/span>[latex] \\frac{15\\sqrt[4]{125}}{\\sqrt[4]{5}} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339436066\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1167339436067\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">34<\/span><span class=\"os-divider\">.<\/span> [latex] 3\\sqrt[3]{-432}+\\sqrt[3]{16} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1167339436110\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Algebraic<\/h3>\r\n<p id=\"fs-id1167339436115\">For the following exercises, simplify each expression.<\/p>\r\n\r\n<div id=\"fs-id1167339436118\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1167339436120\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339436118-solution\">35<\/a><span class=\"os-divider\">.<\/span> [latex] \\sqrt{400x^4} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339436169\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1167339436170\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">36<\/span><span class=\"os-divider\">.<\/span> [latex] \\sqrt{4y^2} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339436197\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1167339436198\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339436197-solution\">37<\/a><span class=\"os-divider\">.<\/span> [latex] \\sqrt{49p} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339436238\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1167339436239\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">38<\/span><span class=\"os-divider\">.<\/span> [latex] \\left(144p^2q^6\\right)^{\\frac{1}{2}} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339436300\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1167339436301\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339436300-solution\">39<\/a><span class=\"os-divider\">.<\/span> [latex] m^{\\frac{5}{2}}\\sqrt{289} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339436367\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1167339436368\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">40<\/span><span class=\"os-divider\">.<\/span> [latex] 9\\sqrt{3m^2}+\\sqrt{27} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339436410\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1167339436411\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339436410-solution\">41<\/a><span class=\"os-divider\">.<\/span> [latex] 3\\sqrt{ab^2}-b\\sqrt{a} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339436472\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1167339436473\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">42<\/span><span class=\"os-divider\">.<\/span> [latex] \\frac{4\\sqrt{2n}}{\\sqrt{16n^4}} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339436524\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1167339436525\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339436524-solution\">43<\/a><span class=\"os-divider\">.<\/span> [latex] \\sqrt{\\frac{225x^3}{49x}} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339436592\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1167339436593\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">44<\/span><span class=\"os-divider\">.<\/span> [latex] 3\\sqrt{44z}+\\sqrt{99z} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339436630\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1167339436632\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339436630-solution\">45<\/a><span class=\"os-divider\">.<\/span> [latex] \\sqrt{50y^8} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339436687\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1167339436688\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">46<\/span><span class=\"os-divider\">.<\/span> [latex] \\sqrt{490bc^2} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339436718\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1167339436719\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339436718-solution\">47<\/a><span class=\"os-divider\">.<\/span> [latex] \\sqrt{\\frac{32}{14d}} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339436792\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1167339436793\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">48<\/span><span class=\"os-divider\">.<\/span> [latex] q^{\\frac{3}{2}}\\sqrt{63p} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339436833\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1167339436834\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339436833-solution\">49<\/a><span class=\"os-divider\">.<\/span> [latex] \\frac{\\sqrt{8}}{1-\\sqrt{3x}} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339436929\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1167339436930\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">50<\/span><span class=\"os-divider\">.<\/span> [latex] \\sqrt{\\frac{20}{121d^4}} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339436971\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1167339436972\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339436971-solution\">51<\/a><span class=\"os-divider\">.<\/span> [latex] w^{\\frac{3}{2}}\\sqrt{32}-w^{\\frac{3}{2}}\\sqrt{50} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339437067\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1167339437068\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">52<\/span><span class=\"os-divider\">.<\/span> [latex] \\sqrt{108x^4}+\\sqrt{27x^4} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339437116\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1167339437117\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339437116-solution\">53<\/a><span class=\"os-divider\">.<\/span> [latex] \\frac{\\sqrt{12x}}{2+2\\sqrt{3}} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339437202\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1167339437203\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">54<\/span><span class=\"os-divider\">.<\/span> [latex] \\sqrt{147k^3} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339437231\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1167339437232\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339437231-solution\">55<\/a><span class=\"os-divider\">.<\/span> [latex] \\sqrt{125n^{10}} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339437291\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1167339437292\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">56<\/span><span class=\"os-divider\">.<\/span> [latex] \\sqrt{\\frac{42q}{36q^3}} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339437334\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1167339437336\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339437334-solution\">57<\/a><span class=\"os-divider\">.<\/span> [latex] \\sqrt{\\frac{81m}{361m^2}} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339437411\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1167339437412\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">58<\/span><span class=\"os-divider\">.<\/span> [latex] \\sqrt{72c}-2\\sqrt{2c} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339437450\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1167339437451\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339437450-solution\">59<\/a><span class=\"os-divider\">.<\/span> [latex] \\sqrt{\\frac{144}{324d^2}} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339437516\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1167339437517\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">60<\/span><span class=\"os-divider\">.<\/span> [latex] \\sqrt[3]{24x^6}+\\sqrt[3]{81x^6} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339437571\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1167339437572\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339437571-solution\">61<\/a><span class=\"os-divider\">.<\/span> [latex] \\sqrt[4]{\\frac{162x^6}{16x^4}} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339437667\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1167339437668\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">62<\/span><span class=\"os-divider\">.<\/span> [latex] \\sqrt[3]{64y} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339437693\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1167339437694\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339437693-solution\">63<\/a><span class=\"os-divider\">.<\/span> [latex] \\sqrt[3]{128z^3}-\\sqrt[3]{-16z^3} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339437774\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1167339437775\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">64<\/span><span class=\"os-divider\">.<\/span> [latex] \\sqrt[5]{1,024c^{10}} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1167339437814\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Real-World Applications<\/h3>\r\n<div id=\"fs-id1167339437820\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1167339437821\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339437820-solution\">65<\/a><span class=\"os-divider\">. <\/span>A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So the length of the guy wire can be found by evaluating [latex] \\sqrt{90,000+160,000}. [\/latex] What is the length of the guy wire?\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339437869\" class=\"material-set-2\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1167339437870\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">66<\/span><span class=\"os-divider\">. <\/span>A car accelerates at a rate of\u00a0[latex] 6 - \\frac{\\sqrt{4}}{\\sqrt{t}} \\, \\text{m\/s}^2 [\/latex] where [latex] t [\/latex] is the time in seconds after the car moves from rest. Simplify the expression.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1167339437932\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Extensions<\/h3>\r\n<p id=\"fs-id1167339437937\">For the following exercises, simplify each expression.<\/p>\r\n\r\n<div id=\"fs-id1167339437940\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1167339437942\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339437940-solution\">67<\/a><span class=\"os-divider\">.<\/span> [latex] \\frac{\\sqrt{8}-\\sqrt{16}}{4-\\sqrt{2}}-2^{\\frac{1}{2}} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339438046\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1167339438047\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">68<\/span><span class=\"os-divider\">.<\/span> [latex] \\frac{4^{\\frac{3}{2}}-16^{\\frac{3}{2}}}{8^{\\frac{1}{3}}} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339438131\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1167339438132\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339438131-solution\">69<\/a><span class=\"os-divider\">.<\/span> [latex] \\frac{\\sqrt{mn^3}}{a^2\\sqrt{c^{-3}}} \\cdot \\frac{a^{-7}n^{-2}}{\\sqrt{m^2c^4}} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339438318\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1167339438319\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">70<\/span><span class=\"os-divider\">.<\/span> [latex] \\frac{a}{a-\\sqrt{c}} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339438349\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1167339438350\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339438349-solution\">71<\/a><span class=\"os-divider\">.<\/span> [latex] \\frac{x\\sqrt{64y} + 4\\sqrt{y}}{\\sqrt{128y}} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339438444\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1167339438445\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">72<\/span><span class=\"os-divider\">.<\/span> [latex] \\left(\\frac{\\sqrt{250x^2}}{\\sqrt{100b^3}}\\right)\\left(\\frac{7\\sqrt{b}}{\\sqrt{125x}}\\right) [\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1167339438550\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1167339438552\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339438550-solution\">73<\/a><span class=\"os-divider\">.<\/span>[latex] \\sqrt{\\frac{\\sqrt[3]{64} + \\sqrt[4]{256}}{\\sqrt{64} + \\sqrt{256}}} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/section><\/div>","rendered":"<div id=\"main-content\" class=\"MainContent__ContentStyles-sc-6yy1if-0 NnXKu\" tabindex=\"-1\" data-dynamic-style=\"true\">\n<div id=\"page_1834bc64-7094-4df1-a530-b3166a295697\" class=\"chapter-content-module\" data-type=\"page\" data-book-content=\"true\">\n<div class=\"textbox textbox--learning-objectives\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Learning Objectives<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>In this section, you will:<\/p>\n<ul>\n<li>Evaluate square roots.<\/li>\n<li>Use the product rule to simplify square roots.<\/li>\n<li>Use the quotient rule to simplify square roots.<\/li>\n<li>Add and subtract square roots.<\/li>\n<li>Rationalize denominators.<\/li>\n<li>Use rational roots.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167339431492\">A hardware store sells 16-ft ladders and 24-ft ladders. A window is located 12 feet above the ground. A ladder needs to be purchased that will reach the window from a point on the ground 5 feet from the building. To find out the length of ladder needed, we can draw a right triangle as shown in Figure 1, and use the Pythagorean Theorem.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_1430\" aria-describedby=\"caption-attachment-1430\" style=\"width: 219px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-1430\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.3-fig-1.webp\" alt=\"\" width=\"219\" height=\"297\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.3-fig-1.webp 219w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.3-fig-1-65x88.webp 65w\" sizes=\"auto, (max-width: 219px) 100vw, 219px\" \/><figcaption id=\"caption-attachment-1430\" class=\"wp-caption-text\">Figure 1<\/figcaption><\/figure>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl}a^2 + b^2 &=& c^2 \\\\5^2 + 12^2 &=& c^2 \\\\169 &=& c^2\\end{array}[\/latex]<\/p>\n<div class=\"chapter-content-module\" data-type=\"page\" data-book-content=\"true\">\n<p id=\"fs-id1167339431629\">Now, we need to find out the length that, when squared, is 169, to determine which ladder to choose. In other words, we need to find a square root. In this section, we will investigate methods of finding solutions to problems such as this one.<\/p>\n<section id=\"fs-id1167339431634\" data-depth=\"1\">\n<h2 data-type=\"title\">Evaluating Square Roots<\/h2>\n<p id=\"fs-id1167339431640\">When the square root of a number is squared, the result is the original number. Since [latex]4^2=16,[\/latex] the square root of [latex]16[\/latex] is [latex]4.[\/latex] The square root function is the inverse of the squaring function just as subtraction is the inverse of addition. To undo squaring, we take the square root.<\/p>\n<p id=\"fs-id1484606\">In general terms, if [latex]a[\/latex] is a positive real number, then the square root of [latex]a[\/latex] is a number that, when multiplied by itself, gives [latex]a.[\/latex] The square root could be positive or negative because multiplying two negative numbers gives a positive number. The <strong><span id=\"term-00006\" data-type=\"term\">principal square root<\/span><\/strong> is the nonnegative number that when multiplied by itself equals [latex]a.[\/latex] The square root obtained using a calculator is the principal square root.<\/p>\n<p id=\"fs-id1507362\">The principal square root of [latex]a[\/latex] is written as [latex]\\sqrt{a}.[\/latex] The symbol is called a <strong><span id=\"term-00007\" data-type=\"term\">radical<\/span><\/strong>, the term under the symbol is called the <strong><span id=\"term-00008\" data-type=\"term\">radicand<\/span><\/strong>, and the entire expression is called a <strong><span id=\"term-00009\" data-type=\"term\">radical expression<\/span><\/strong>.<span id=\"fs-id1495127\" data-type=\"media\" data-alt=\"The expression: square root of twenty-five is enclosed in a circle. The circle has an arrow pointing to it labeled: Radical expression. The square root symbol has an arrow pointing to it labeled: Radical. The number twenty-five has an arrow pointing to it labeled: Radicand.\"><\/span><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-1431 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.3-rad-300x129.webp\" alt=\"\" width=\"300\" height=\"129\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.3-rad-300x129.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.3-rad-65x28.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.3-rad-225x97.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.3-rad.webp 311w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Principal Square Root<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>The <span id=\"term-00010\" data-type=\"term\">principal square root<\/span> of\u00a0[latex]a[\/latex] is the nonnegative number that, when multiplied by itself, equals\u00a0[latex]a.[\/latex] It is written as a <span id=\"term-00011\" data-type=\"term\">radical expression<\/span><strong>,<\/strong> with a symbol called a <span id=\"term-00012\" data-type=\"term\">radical<\/span> over the term called the <span id=\"term-00013\" data-type=\"term\">radicand<\/span>: [latex]\\sqrt{a}.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Q&amp;A<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Q: Does<\/strong> [latex]\\sqrt{25}=\\pm5?[\/latex]<\/p>\n<p><em>A: No. Although both\u00a0[latex]5^2[\/latex] and\u00a0[latex](-5)^2[\/latex] are\u00a0[latex]25,[\/latex] the radical symbol implies only a nonnegative root, the principal square root. The principal square root of 25 is<\/em> [latex]\\sqrt{25}=5.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 1: Evaluating Square Roots<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Evaluate each expression.<\/p>\n<p>(a) [latex]\\sqrt{100}[\/latex]<\/p>\n<p>(b) [latex]\\sqrt{\\sqrt{16}}[\/latex]<\/p>\n<p>(c) [latex]\\sqrt{25+144}[\/latex]<\/p>\n<p>(d) [latex]\\sqrt{49}-\\sqrt{81}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>(a) [latex]\\sqrt{100}=10[\/latex] because [latex]10^2=100[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>(b) [latex]\\sqrt{\\sqrt{16}}=\\sqrt{4}[\/latex] because [latex]4^2=16[\/latex] and [latex]2^2=4[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>(c) [latex]\\sqrt{25+144}=\\sqrt{169}=13[\/latex] because [latex]13^2=169[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>(d) [latex]\\sqrt{49}-\\sqrt{81}=7-9=-2[\/latex] because [latex]7^2=49[\/latex] and [latex]9^2=81[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Q&amp;A<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Q: For\u00a0[latex]\\sqrt{25+144},[\/latex] can we find the square roots before adding?<\/strong><\/p>\n<p><em>A: No.\u00a0[latex]\\sqrt{25}+\\sqrt{144}=5+12=17.[\/latex] This is not equivalent to\u00a0[latex]\\sqrt{25+144}=13.[\/latex] The order of operations requires us to add the terms in the radicand before finding the square root.<\/em><\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Evaluate each expression.<\/p>\n<p>(a) [latex]\\sqrt{225}[\/latex]<\/p>\n<p>(b) [latex]\\sqrt{\\sqrt{81}}[\/latex]<\/p>\n<p>(c) [latex]\\sqrt{25-9}[\/latex]<\/p>\n<p>(d) [latex]\\sqrt{36}+\\sqrt{121}[\/latex]<\/p>\n<\/div>\n<\/div>\n<section id=\"fs-id1424126\" data-depth=\"1\">\n<h2 data-type=\"title\">Using the Product Rule to Simplify Square Roots<\/h2>\n<p id=\"fs-id1499813\">To simplify a square root, we rewrite it such that there are no perfect squares in the radicand. There are several properties of square roots that allow us to simplify complicated radical expressions. The first rule we will look at is the <em data-effect=\"italics\">product rule for simplifying square roots,<\/em> which allows us to separate the square root of a product of two numbers into the product of two separate rational expressions. For instance, we can rewrite [latex]\\sqrt{15}[\/latex] as [latex]\\sqrt{3}\\cdot \\sqrt{5}[\/latex] We can also use the product rule to express the product of multiple radical expressions as a single radical expression.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">The Product Rule for Simplifying Square Roots<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"eip-id1167336040491\">If [latex]a[\/latex] and [latex]b[\/latex] are nonnegative, the square root of the product [latex]ab[\/latex] is equal to the product of the square roots of [latex]a[\/latex] and [latex]b.[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt{ab}=\\sqrt{a}\\cdot \\sqrt{b}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How To<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id1343706\"><strong>Given a square root radical expression, use the product rule to simplify it.<br \/>\n<\/strong><\/p>\n<ol id=\"fs-id1516957\" type=\"1\">\n<li>Factor any perfect squares from the radicand.<\/li>\n<li>Write the radical expression as a product of radical expressions.<\/li>\n<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 2: Using the Product Rule to Simplify Square Roots<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Simplify the radical expression.<\/p>\n<p>(a) [latex]\\sqrt{300}[\/latex]<\/p>\n<p>(b) [latex]\\sqrt{162a^5b^4}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>(a)<\/p>\n<p>[latex]\\begin{array}{ll}\\sqrt{100 \\cdot 3} & \\quad \\text{Factor perfect square from radicand.} \\\\\\sqrt{100} \\cdot \\sqrt{3} & \\quad \\text{Write radical expression as product of radical expressions.} \\\\10\\sqrt{3} & \\quad \\text{Simplify.}\\end{array}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>(b)<\/p>\n<p>[latex]\\begin{array}{ll}\\sqrt{81a^4b^4 \\cdot 2a} & \\quad \\text{Factor perfect square from radicand.} \\\\\\sqrt{81a^4b^4} \\cdot \\sqrt{2a} & \\quad \\text{Write radical expression as product of radical expressions.} \\\\9a^2b^2\\sqrt{2a} & \\quad \\text{Simplify.}\\end{array}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/section>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Simplify [latex]\\sqrt{50x^2y^3z}.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How To<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id537400\"><strong>Given the product of multiple radical expressions, use the product rule to combine them into one radical expression.<br \/>\n<\/strong><\/p>\n<ol id=\"fs-id1343311\" type=\"1\">\n<li>Express the product of multiple radical expressions as a single radical expression.<\/li>\n<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 3: Using the Product Rule to Simplify the Product of Multiple Square Roots<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Simplify the radical expression.<\/p>\n<p>[latex]\\sqrt{12}\\cdot \\sqrt{3}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll}\\sqrt{12 \\cdot 3} & \\quad \\text{Express the product as a single radical expression.} \\\\\\sqrt{36} & \\quad \\text{Simplify.} \\\\6 & \\end{array}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #3<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Simplify [latex]\\sqrt{50x}\\cdot \\sqrt{2x}[\/latex] assuming [latex]x> 0.[\/latex]<\/p>\n<\/div>\n<\/div>\n<section id=\"fs-id1409429\" data-depth=\"1\">\n<h2 data-type=\"title\">Using the Quotient Rule to Simplify Square Roots<\/h2>\n<p id=\"fs-id1402495\">Just as we can rewrite the square root of a product as a product of square roots, so too can we rewrite the square root of a quotient as a quotient of square roots, using the <em data-effect=\"italics\">quotient rule for simplifying square roots.<\/em> It can be helpful to separate the numerator and denominator of a fraction under a radical so that we can take their square roots separately. We can rewrite [latex]\\sqrt{\\frac{5}{2}}[\/latex] as [latex]\\frac{\\sqrt{5}}{\\sqrt{2}}.[\/latex]<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">The Quotient Rule for Simplifying Square Roots<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>The square root of the quotient [latex]\\frac{a}{b}[\/latex] is equal to the quotient of the square roots of [latex]a[\/latex] and [latex]b,[\/latex] where [latex]b\\not= 0.[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt{\\frac{a}{b}}=\\frac{\\sqrt{a}}{\\sqrt{b}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How To<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id1548985\"><strong>Given a radical expression, use the quotient rule to simplify it.<br \/>\n<\/strong><\/p>\n<ol id=\"fs-id1537621\" type=\"1\">\n<li>Write the radical expression as the quotient of two radical expressions.<\/li>\n<li>Simplify the numerator and denominator.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 4: Using the Quotient Rule to Simplify Square Roots<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Simplify the radical expression.<\/p>\n<p>[latex]\\sqrt{\\frac{5}{36}}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll}\\frac{\\sqrt{5}}{\\sqrt{36}} & \\quad \\text{Write as a quotient of two radical expressions.} \\\\ \\frac{\\sqrt{5}}{6} & \\quad \\text{Simplify denominator.} \\end{array}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/section>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #4<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Simplify [latex]\\sqrt{\\frac{2x^2}{9y^4}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 5: Using the Quotient Rule to Simplify an Expression with Two Square Roots<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Simplify the radical expression.<\/p>\n<p>[latex]\\frac{\\sqrt{234x^{11}y}}{\\sqrt{26x^7y}}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p style=\"text-align: left;\">[latex]\\begin{array}{ll}\\sqrt{\\frac{234x^{11}y}{26x^7y}} & \\quad \\text{Combine numerator and denominator into one radical expression.} \\\\ \\sqrt{9x^4} & \\quad \\text{Simplify fraction.} \\\\ 3x^2 & \\quad \\text{Simplify square root.} \\end{array}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #5<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Simplify [latex]\u00a0 \\frac{\\sqrt{9a^5b^{14}}}{\\sqrt{3a^4b^5}}.[\/latex]<\/p>\n<\/div>\n<\/div>\n<section id=\"fs-id1413677\" data-depth=\"1\">\n<h2 data-type=\"title\">Adding and Subtracting Square Roots<\/h2>\n<p id=\"fs-id1483382\">We can add or subtract radical expressions only when they have the same radicand and when they have the same radical type such as square roots. For example, the sum of [latex]\\sqrt{2}[\/latex] and [latex]3\\sqrt{2}[\/latex] is [latex]4\\sqrt{2}.[\/latex] However, it is often possible to simplify radical expressions, and that may change the radicand. The radical expression [latex]\\sqrt{18}[\/latex] can be written with a [latex]2[\/latex] in the radicand, as [latex]3\\sqrt{2},[\/latex] so [latex]\\sqrt{2}+\\sqrt{18}=\\sqrt{2}+3\\sqrt{2}=4\\sqrt{2}.[\/latex]<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How To<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id1353954\"><strong>Given a radical expression requiring addition or subtraction of square roots, simplify.<\/strong><\/p>\n<ol id=\"fs-id1429237\" type=\"1\">\n<li>Simplify each radical expression.<\/li>\n<li>Add or subtract expressions with equal radicands.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 6: Adding Square Roots<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Add [latex]5\\sqrt{12}+2\\sqrt{3}.[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p id=\"fs-id1537392\">We can rewrite [latex]5\\sqrt{12}[\/latex] as [latex]5\\sqrt{4\\cdot 3}.[\/latex] According the product rule, this becomes [latex]5\\sqrt{4}\\sqrt{3}.[\/latex] The square root of [latex]\\sqrt{4}[\/latex] is 2, so the expression becomes [latex]5(2)\\sqrt{3},[\/latex] which is [latex]10\\sqrt{3}.[\/latex] Now the terms have the same radicand so we can add.<\/p>\n<p style=\"text-align: center;\">[latex]10\\sqrt{3}+2\\sqrt{3}=12\\sqrt{3}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/section>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #6<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Add [latex]\\sqrt{5}+6\\sqrt{20}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 7: Subtracting Square Roots<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Subtract [latex]20\\sqrt{72a^3b^4c}-14\\sqrt{8a^3b^4c}.[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>Factor 9 out of the first term so that both terms have equal radicands.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rclcl}20\\sqrt{72a^3b^4c} &=& 20\\sqrt{9 \\cdot 8a^3b^4c} &=& 20\\sqrt{9}\\sqrt{8a^3b^4c} \\\\&=& 20(3)\\sqrt{8a^3b^4c} &=& 60\\sqrt{8a^3b^4c}\\end{array}[\/latex]<\/p>\n<p>So<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}20\\sqrt{72a^3b^4c} - 14\\sqrt{8a^3b^4c} \\\\= 60\\sqrt{8a^3b^4c} - 14\\sqrt{8a^3b^4c} \\\\= 46\\sqrt{8a^3b^4c}\\end{array}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #7<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Subtract [latex]3\\sqrt{80x}-4\\sqrt{45x}.[\/latex]<\/p>\n<\/div>\n<\/div>\n<section id=\"fs-id1483223\" data-depth=\"1\">\n<h2 data-type=\"title\">Rationalizing Denominators<\/h2>\n<p id=\"fs-id1483228\">When an expression involving square root radicals is written in simplest form, it will not contain a radical in the denominator. We can remove radicals from the denominators of fractions using a process called <em data-effect=\"italics\">rationalizing the denominator<\/em>.<\/p>\n<p id=\"fs-id1517983\">We know that multiplying by 1 does not change the value of an expression. We use this property of multiplication to change expressions that contain radicals in the denominator. To remove radicals from the denominators of fractions, multiply by the form of 1 that will eliminate the radical.<\/p>\n<p id=\"fs-id1517989\">For a denominator containing a single term, multiply by the radical in the denominator over itself. In other words, if the denominator is [latex]b\\sqrt{c},[\/latex] multiply by [latex]\\frac{\\sqrt{c}}{\\sqrt{c}}.[\/latex]<\/p>\n<p id=\"fs-id1484146\">For a denominator containing the sum or difference of a rational and an irrational term, multiply the numerator and denominator by the conjugate of the denominator, which is found by changing the sign of the radical portion of the denominator. If the denominator is [latex]a+b\\sqrt{c},[\/latex] then the conjugate is [latex]a-b\\sqrt{c}.[\/latex]<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How To<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id1507281\"><strong>Given an expression with a single square root radical term in the denominator, rationalize the denominator.<\/strong><\/p>\n<ol id=\"fs-id1507286\" type=\"a\">\n<li>Multiply the numerator and denominator by the radical in the denominator.<\/li>\n<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 8: Rationalizing a Denominator Containing a Single Term<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Write [latex]\\frac{2\\sqrt{3}}{3\\sqrt{10}}[\/latex] in simplest form.<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>The radical in the denominator is [latex]\\sqrt{10}.[\/latex] So multiply the fraction by [latex]\\frac{\\sqrt{10}}{\\sqrt{10}}.[\/latex] Then simplify.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\frac{2\\sqrt{3}}{3\\sqrt{10}} \\cdot \\frac{\\sqrt{10}}{\\sqrt{10}} \\quad \\\\\\frac{2\\sqrt{30}}{30} \\quad \\\\\\frac{\\sqrt{30}}{15}\\end{array}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/section>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #8<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Write [latex]\\frac{12\\sqrt{3}}{\\sqrt{2}}[\/latex] in simplest form.<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How To<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id1167339431931\"><strong>Given an expression with a radical term and a constant in the denominator, rationalize the denominator.<\/strong><\/p>\n<ol id=\"fs-id1167339431936\" type=\"1\">\n<li>Find the conjugate of the denominator.<\/li>\n<li>Multiply the numerator and denominator by the conjugate.<\/li>\n<li>Use the distributive property.<\/li>\n<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 9: Rationalizing a Denominator Containing Two Terms<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Write [latex]\\frac{4}{1+\\sqrt{5}}[\/latex] in simplest form.<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>Begin by finding the conjugate of the denominator by writing the denominator and changing the sign. So the conjugate of [latex]1+\\sqrt{5}[\/latex] is [latex]1-\\sqrt{5}.[\/latex] Then multiply the fraction by [latex]\\frac{1-\\sqrt{5}}{1-\\sqrt{5}}.[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll}\\frac{4}{1 + \\sqrt{5}} \\cdot \\frac{1 - \\sqrt{5}}{1 - \\sqrt{5}} & \\\\\\frac{4 - 4\\sqrt{5}}{-4} & \\quad \\text{Use the distributive property.} \\\\\\sqrt{5} - 1 & \\quad \\text{Simplify.}\\end{array}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #9<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Write [latex]\\frac{7}{2+\\sqrt{3}}[\/latex] in simplest form.<\/p>\n<\/div>\n<\/div>\n<section id=\"fs-id1167339432042\" data-depth=\"1\">\n<h2 data-type=\"title\">Using Rational Roots<\/h2>\n<p id=\"fs-id1167339432047\">Although square roots are the most common rational roots, we can also find cube roots, 4th roots, 5th roots, and more. Just as the square root function is the inverse of the squaring function, these roots are the inverse of their respective power functions. These functions can be useful when we need to determine the number that, when raised to a certain power, gives a certain number.<\/p>\n<section id=\"fs-id1167339432053\" data-depth=\"2\">\n<h3 data-type=\"title\">Understanding <em data-effect=\"italics\">n<\/em>th Roots<\/h3>\n<p id=\"fs-id1167339432064\">Suppose we know that [latex]a^3=8.[\/latex] We want to find what number raised to the 3rd power is equal to 8. Since [latex]2^3=8,[\/latex] we say that 2 is the cube root of 8.<\/p>\n<p id=\"fs-id1167339432119\">The <em data-effect=\"italics\">n<\/em>th root of [latex]a[\/latex] is a number that, when raised to the <em data-effect=\"italics\">n<\/em>th power, gives [latex]a.[\/latex] For example, [latex]-3[\/latex] is the 5th root of [latex]-243[\/latex] because [latex](-3)^5=-243.[\/latex] If [latex]a[\/latex] is a real number with at least one <em data-effect=\"italics\">n<\/em>th root, then the <strong>principal <em data-effect=\"italics\">n<\/em>th root<\/strong> of [latex]a[\/latex] is the number with the same sign as [latex]a[\/latex] that, when raised to the <em data-effect=\"italics\">n<\/em>th power, equals [latex]a.[\/latex]<\/p>\n<p id=\"fs-id1167339434352\">The principal <em data-effect=\"italics\">n<\/em>th root of [latex]a[\/latex] is written as [latex]\\sqrt[n]{a},[\/latex] where [latex]n[\/latex] is a positive integer greater than or equal to 2. In the radical expression, [latex]n[\/latex] is called the <strong>index<\/strong> of the radical.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Principal <em>n<\/em>th Root<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id1167339434411\">If [latex]a[\/latex] is a real number with at least one <em data-effect=\"italics\">n<\/em>th root, then the <span id=\"term-00016\" data-type=\"term\"><strong>principal<\/strong> <em data-effect=\"italics\">n<\/em>th root<\/span> of [latex]a,[\/latex] written as [latex]\\sqrt[n]{a},[\/latex] is the number with the same sign as [latex]a[\/latex] that, when raised to the <em data-effect=\"italics\">n<\/em>th power, equals [latex]a.[\/latex] The <strong>index <\/strong>of the radical is [latex]n.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/section>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 10: Simplifying <em>n<\/em>th Roots<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Simplify each of the following:<\/p>\n<p>(a) [latex]\\sqrt[5]{-32}[\/latex]<\/p>\n<p>(b) [latex]\\sqrt[4]{4}\\cdot \\sqrt[4]{1,024}[\/latex]<\/p>\n<p>(c) [latex]-\\sqrt[3]{\\frac{8x^6}{125}}[\/latex]<\/p>\n<p>(d) [latex]8\\sqrt[4]{3}-\\sqrt[4]{48}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>(a) [latex]\\sqrt[5]{-32}=-2[\/latex] because [latex](-2)^5=-32[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>(b) First, express the product as a single radical expression. because [latex]\\sqrt[4]{4,096}=8[\/latex] because [latex]8^4=4,096[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>(c)<\/p>\n<p>[latex]\\begin{array}{ll}\\frac{-\\sqrt[3]{8x^6}}{\\sqrt[3]{125}} & \\quad \\text{Write as quotient of two radical expressions.} \\\\\\frac{-2x^2}{5} & \\quad \\text{Simplify.}\\end{array}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>(d)<\/p>\n<p>[latex]\\begin{array}{ll}8\\sqrt[4]{3} - 2\\sqrt[4]{3} & \\quad \\text{Simplify to get equal radicands.} \\\\6\\sqrt[4]{3} & \\quad \\text{Add.}\\end{array}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #10<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Simplify.<\/p>\n<p>(a) [latex]\\sqrt[3]{-216}[\/latex]<\/p>\n<p>(b) [latex]\\frac{3\\sqrt[4]{80}}{\\sqrt[4]{5}}[\/latex]<\/p>\n<p>(c) [latex]6\\sqrt[3]{9,000}+7\\sqrt[3]{576}[\/latex]<\/p>\n<\/div>\n<\/div>\n<section id=\"fs-id1538427\" data-depth=\"2\">\n<h3 data-type=\"title\">Using Rational Exponents<\/h3>\n<p id=\"fs-id1538433\"><strong>Radical expressions <\/strong>can also be written without using the radical symbol. We can use rational (fractional) exponents. The index must be a positive integer. If the index [latex]n[\/latex] is even, then [latex]a[\/latex] cannot be negative.<\/p>\n<p style=\"text-align: center;\">[latex]a^{\\frac{1}{n}}=\\sqrt[n]{a}[\/latex]<\/p>\n<p id=\"fs-id1508564\">We can also have rational exponents with numerators other than 1. In these cases, the exponent must be a fraction in lowest terms. We raise the base to a power and take an <em data-effect=\"italics\">n<\/em>th root. The numerator tells us the power and the denominator tells us the root.<\/p>\n<p style=\"text-align: center;\">[latex]a^{\\frac{m}{n}}=\\left(\\sqrt[n]{a}\\right)^m=\\sqrt[n]{a^m}[\/latex]<\/p>\n<p id=\"fs-id1529202\">All of the properties of exponents that we learned for integer exponents also hold for rational exponents.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Rational Exponents<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Rational exponents are another way to express principal <em data-effect=\"italics\">n<\/em>th roots. The general form for converting between a radical expression with a radical symbol and one with a rational exponent is<\/p>\n<p style=\"text-align: center;\">[latex]a^{\\frac{m}{n}}=\\left(\\sqrt[n]{a}\\right)^m=\\sqrt[n]{a^m}\u00a0[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How To<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"eip-id1592533\"><strong>Given an expression with a rational exponent, write the expression as a radical.<\/strong><\/p>\n<ol id=\"fs-id1482550\" type=\"1\">\n<li>Determine the power by looking at the numerator of the exponent.<\/li>\n<li>Determine the root by looking at the denominator of the exponent.<\/li>\n<li>Using the base as the radicand, raise the radicand to the power and use the root as the index.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 11: Writing Rational Exponents as Radicals<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Write [latex]343^{\\frac{2}{3}}[\/latex] as a radical. Simplify.<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>The 2 tells us the power and the 3 tells us the root.<\/p>\n<p style=\"text-align: center;\">[latex]343^{\\frac{2}{3}}=\\left(\\sqrt[3]{343}\\right)^2=\\sqrt[3]{343^2}[\/latex]<\/p>\n<p>We know that [latex]\\sqrt[3]{343}=7[\/latex] because [latex]7^3=343.[\/latex] Because the cube root is easy to find, it is easiest to find the cube root before squaring for this problem. In general, it is easier to find the root first and then raise it to a power.<\/p>\n<p style=\"text-align: center;\">[latex]343^{\\frac{2}{3}}=\\left(\\sqrt[3]{343}\\right)^2=7^2=49[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/section>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #11<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Write [latex]9^{\\frac{5}{2}}[\/latex] as a radical. Simplify.<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 12: Writing Radicals as Rational Exponents<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Write [latex]\\frac{4}{\\sqrt[7]{a^2}}[\/latex] using a rational exponent.<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>The power is 2 and the root is 7, so the rational exponent will be [latex]\\frac{2}{7}.[\/latex] We get [latex]\\frac{4}{a^{\\frac{2}{7}}}.[\/latex] Using properties of exponents, we get [latex]\\frac{4}{\\sqrt[7]{a^2}}=4a^{\\frac{-2}{7}}.[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #12<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Write [latex]x\\sqrt{(5y)^9}[\/latex] using a rational exponent.<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 13: Simplifying Rational Exponents<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Simplify:<\/p>\n<p>(a) [latex]5\\left(2x^{\\frac{3}{4}}\\right)\\left(3x^{\\frac{1}{5}}\\right)[\/latex]<\/p>\n<p>(b) [latex]\\left(\\frac{16}{9}\\right)^{-\\frac{1}{2}}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>(a)<\/p>\n<p>[latex]\\begin{array}{ll}30x^{\\frac{3}{4}}x^{\\frac{1}{5}} & \\quad \\text{Multiply the coefficients.} \\\\30x^{\\frac{3}{4} + \\frac{1}{5}} & \\quad \\text{Use properties of exponents.} \\\\30x^{\\frac{19}{20}} & \\quad \\text{Simplify.}\\end{array}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>(b)<\/p>\n<p>[latex]\\begin{array}{ll} \\left(\\frac{9}{16}\\right)^{\\frac{1}{2}} & \\quad \\text{ Use definition of negative exponents.} \\\\\\sqrt{\\frac{9}{16}} & \\quad \\text{ Rewrite as a radical.} \\\\\\frac{\\sqrt{9}}{\\sqrt{16}} & \\quad \\text{ Use the quotient rule.} \\\\\\frac{3}{4} & \\quad \\text{ Simplify.}\\end{array}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #13<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Simplify [latex](8x)^{\\frac{1}{3}}\\left(14x^{\\frac{6}{5}}\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Media<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id1487284\">Access these online resources for additional instruction and practice with radicals and rational exponents.<\/p>\n<ul id=\"fs-id1476938\">\n<li><a href=\"http:\/\/openstax.org\/l\/introradical\" target=\"_blank\" rel=\"noopener nofollow\">Radicals<\/a><\/li>\n<li><a href=\"http:\/\/openstax.org\/l\/rationexpon\" target=\"_blank\" rel=\"noopener nofollow\">Rational Exponents<\/a><\/li>\n<li><a href=\"http:\/\/openstax.org\/l\/simpradical\" target=\"_blank\" rel=\"noopener nofollow\">Simplify Radicals<\/a><\/li>\n<li><a href=\"http:\/\/openstax.org\/l\/rationdenom\" target=\"_blank\" rel=\"noopener nofollow\">Rationalize Denominator<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\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\">1.3 Section Exercises<\/span><\/h2>\n<section id=\"fs-id1499197\" class=\"section-exercises\" data-depth=\"1\">\n<section id=\"fs-id1499203\" data-depth=\"2\">\n<h3 data-type=\"title\">Verbal<\/h3>\n<div id=\"fs-id1499209\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1499210\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1499209-solution\">1<\/a><span class=\"os-divider\">. <\/span>What does it mean when a radical does not have an index? Is the expression equal to the radicand? Explain.<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1538359\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1538360\" data-type=\"problem\">\n<p><span class=\"os-number\">2<\/span><span class=\"os-divider\">. <\/span>Where would radicals come in the order of operations? Explain why.<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1538365\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1538366\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1538365-solution\">3<\/a><span class=\"os-divider\">. <\/span>Every number will have two square roots. What is the principal square root?<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1538374\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1538375\" data-type=\"problem\">\n<p><span class=\"os-number\">4<\/span><span class=\"os-divider\">. <\/span>Can a radical with a negative radicand have a real square root? Why or why not?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1538381\" data-depth=\"2\">\n<h3 data-type=\"title\">Numeric<\/h3>\n<p id=\"fs-id1538386\">For the following exercises, simplify each expression.<\/p>\n<div id=\"fs-id1514811\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1514812\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1514811-solution\">5<\/a><span class=\"os-divider\">.<\/span> [latex]\\sqrt{256}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1514836\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1514837\" data-type=\"problem\">\n<p><span class=\"os-number\">6<\/span><span class=\"os-divider\">.<\/span> [latex]\\sqrt{\\sqrt{256}}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1499788\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1499789\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1499788-solution\">7<\/a><span class=\"os-divider\">.<\/span> [latex]\\sqrt{4(9+16)}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1549261\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1549262\" data-type=\"problem\">\n<p><span class=\"os-number\">8<\/span><span class=\"os-divider\">.<\/span> [latex]\\sqrt{289}-\\sqrt{121}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1529781\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1529782\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1529781-solution\">9<\/a><span class=\"os-divider\">.<\/span> [latex]\\sqrt{196}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1430084\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1430085\" data-type=\"problem\">\n<p><span class=\"os-number\">10<\/span><span class=\"os-divider\">.<\/span> [latex]\\sqrt{1}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1430101\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1430102\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1430101-solution\">11<\/a><span class=\"os-divider\">.<\/span> [latex]\\sqrt{98}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1487491\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1487492\" data-type=\"problem\">\n<p><span class=\"os-number\">12<\/span><span class=\"os-divider\">.<\/span> [latex]\\sqrt{\\frac{27}{64}}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1545152\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1545153\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1545152-solution\">13<\/a><span class=\"os-divider\">.<\/span> [latex]\\sqrt{\\frac{81}{5}}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1469422\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1469424\" data-type=\"problem\">\n<p><span class=\"os-number\">14<\/span><span class=\"os-divider\">.<\/span> [latex]\\sqrt{800}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1469443\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1469444\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1469443-solution\">15<\/a><span class=\"os-divider\">.<\/span> [latex]\\sqrt{169}+\\sqrt{144}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339435248\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1167339435249\" data-type=\"problem\">\n<p><span class=\"os-number\">16<\/span><span class=\"os-divider\">.<\/span> [latex]\\sqrt{\\frac{8}{50}}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339435278\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1167339435279\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339435278-solution\">17<\/a><span class=\"os-divider\">.<\/span> [latex]\\frac{18}{\\sqrt{162}}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339435349\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1167339435350\" data-type=\"problem\">\n<p><span class=\"os-number\">18<\/span><span class=\"os-divider\">.<\/span> [latex]\\sqrt{192}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339435370\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1167339435371\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339435370-solution\">19<\/a><span class=\"os-divider\">.<\/span> [latex]14\\sqrt{6}-6\\sqrt{24}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339435421\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1167339435422\" data-type=\"problem\">\n<p><span class=\"os-number\">20<\/span><span class=\"os-divider\">.<\/span> [latex]15\\sqrt{5}+7\\sqrt{45}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339435454\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1167339435455\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339435454-solution\">21<\/a><span class=\"os-divider\">.<\/span> [latex]\\sqrt{150}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339435492\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1167339435493\" data-type=\"problem\">\n<p><span class=\"os-number\">22<\/span><span class=\"os-divider\">.<\/span> [latex]\\sqrt{\\frac{96}{100}}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339435525\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1167339435526\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339435525-solution\">23<\/a><span class=\"os-divider\">.<\/span> [latex]\\left(\\sqrt{42}\\right) \\left(\\sqrt{30}\\right)[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339435597\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1167339435598\" data-type=\"problem\">\n<p><span class=\"os-number\">24<\/span><span class=\"os-divider\">.<\/span> [latex]12\\sqrt{3}-4\\sqrt{75}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339435630\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1167339435631\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339435630-solution\">25<\/a><span class=\"os-divider\">.<\/span> [latex]\\sqrt{\\frac{4}{225}}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339435682\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1167339435683\" data-type=\"problem\">\n<p><span class=\"os-number\">26<\/span><span class=\"os-divider\">.<\/span> [latex]\\sqrt{\\frac{405}{324}}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339435715\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1167339435716\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339435715-solution\">27<\/a><span class=\"os-divider\">.<\/span> [latex]\\sqrt{\\frac{360}{361}}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339435782\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1167339435784\" data-type=\"problem\">\n<p><span class=\"os-number\">28<\/span><span class=\"os-divider\">.<\/span> [latex]\\frac{5}{1+\\sqrt{3}}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339435813\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1167339435814\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339435813-solution\">29<\/a><span class=\"os-divider\">.<\/span> [latex]\\frac{8}{1-\\sqrt{17}}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339435883\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1167339435884\" data-type=\"problem\">\n<p><span class=\"os-number\">30<\/span><span class=\"os-divider\">.<\/span> [latex]\\sqrt[4]{16}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339435906\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1167339435907\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339435906-solution\">31<\/a><span class=\"os-divider\">.<\/span> [latex]\\sqrt[3]{128}+3\\sqrt[3]{2}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339435965\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1167339435966\" data-type=\"problem\">\n<p><span class=\"os-number\">32<\/span><span class=\"os-divider\">.<\/span> [latex]\\sqrt[5]{\\frac{-32}{243}}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339436003\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1167339436004\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339436003-solution\">33<\/a><span class=\"os-divider\">.<\/span>[latex]\\frac{15\\sqrt[4]{125}}{\\sqrt[4]{5}}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339436066\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1167339436067\" data-type=\"problem\">\n<p><span class=\"os-number\">34<\/span><span class=\"os-divider\">.<\/span> [latex]3\\sqrt[3]{-432}+\\sqrt[3]{16}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1167339436110\" data-depth=\"2\">\n<h3 data-type=\"title\">Algebraic<\/h3>\n<p id=\"fs-id1167339436115\">For the following exercises, simplify each expression.<\/p>\n<div id=\"fs-id1167339436118\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1167339436120\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339436118-solution\">35<\/a><span class=\"os-divider\">.<\/span> [latex]\\sqrt{400x^4}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339436169\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1167339436170\" data-type=\"problem\">\n<p><span class=\"os-number\">36<\/span><span class=\"os-divider\">.<\/span> [latex]\\sqrt{4y^2}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339436197\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1167339436198\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339436197-solution\">37<\/a><span class=\"os-divider\">.<\/span> [latex]\\sqrt{49p}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339436238\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1167339436239\" data-type=\"problem\">\n<p><span class=\"os-number\">38<\/span><span class=\"os-divider\">.<\/span> [latex]\\left(144p^2q^6\\right)^{\\frac{1}{2}}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339436300\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1167339436301\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339436300-solution\">39<\/a><span class=\"os-divider\">.<\/span> [latex]m^{\\frac{5}{2}}\\sqrt{289}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339436367\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1167339436368\" data-type=\"problem\">\n<p><span class=\"os-number\">40<\/span><span class=\"os-divider\">.<\/span> [latex]9\\sqrt{3m^2}+\\sqrt{27}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339436410\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1167339436411\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339436410-solution\">41<\/a><span class=\"os-divider\">.<\/span> [latex]3\\sqrt{ab^2}-b\\sqrt{a}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339436472\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1167339436473\" data-type=\"problem\">\n<p><span class=\"os-number\">42<\/span><span class=\"os-divider\">.<\/span> [latex]\\frac{4\\sqrt{2n}}{\\sqrt{16n^4}}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339436524\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1167339436525\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339436524-solution\">43<\/a><span class=\"os-divider\">.<\/span> [latex]\\sqrt{\\frac{225x^3}{49x}}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339436592\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1167339436593\" data-type=\"problem\">\n<p><span class=\"os-number\">44<\/span><span class=\"os-divider\">.<\/span> [latex]3\\sqrt{44z}+\\sqrt{99z}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339436630\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1167339436632\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339436630-solution\">45<\/a><span class=\"os-divider\">.<\/span> [latex]\\sqrt{50y^8}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339436687\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1167339436688\" data-type=\"problem\">\n<p><span class=\"os-number\">46<\/span><span class=\"os-divider\">.<\/span> [latex]\\sqrt{490bc^2}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339436718\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1167339436719\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339436718-solution\">47<\/a><span class=\"os-divider\">.<\/span> [latex]\\sqrt{\\frac{32}{14d}}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339436792\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1167339436793\" data-type=\"problem\">\n<p><span class=\"os-number\">48<\/span><span class=\"os-divider\">.<\/span> [latex]q^{\\frac{3}{2}}\\sqrt{63p}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339436833\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1167339436834\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339436833-solution\">49<\/a><span class=\"os-divider\">.<\/span> [latex]\\frac{\\sqrt{8}}{1-\\sqrt{3x}}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339436929\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1167339436930\" data-type=\"problem\">\n<p><span class=\"os-number\">50<\/span><span class=\"os-divider\">.<\/span> [latex]\\sqrt{\\frac{20}{121d^4}}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339436971\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1167339436972\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339436971-solution\">51<\/a><span class=\"os-divider\">.<\/span> [latex]w^{\\frac{3}{2}}\\sqrt{32}-w^{\\frac{3}{2}}\\sqrt{50}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339437067\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1167339437068\" data-type=\"problem\">\n<p><span class=\"os-number\">52<\/span><span class=\"os-divider\">.<\/span> [latex]\\sqrt{108x^4}+\\sqrt{27x^4}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339437116\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1167339437117\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339437116-solution\">53<\/a><span class=\"os-divider\">.<\/span> [latex]\\frac{\\sqrt{12x}}{2+2\\sqrt{3}}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339437202\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1167339437203\" data-type=\"problem\">\n<p><span class=\"os-number\">54<\/span><span class=\"os-divider\">.<\/span> [latex]\\sqrt{147k^3}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339437231\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1167339437232\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339437231-solution\">55<\/a><span class=\"os-divider\">.<\/span> [latex]\\sqrt{125n^{10}}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339437291\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1167339437292\" data-type=\"problem\">\n<p><span class=\"os-number\">56<\/span><span class=\"os-divider\">.<\/span> [latex]\\sqrt{\\frac{42q}{36q^3}}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339437334\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1167339437336\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339437334-solution\">57<\/a><span class=\"os-divider\">.<\/span> [latex]\\sqrt{\\frac{81m}{361m^2}}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339437411\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1167339437412\" data-type=\"problem\">\n<p><span class=\"os-number\">58<\/span><span class=\"os-divider\">.<\/span> [latex]\\sqrt{72c}-2\\sqrt{2c}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339437450\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1167339437451\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339437450-solution\">59<\/a><span class=\"os-divider\">.<\/span> [latex]\\sqrt{\\frac{144}{324d^2}}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339437516\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1167339437517\" data-type=\"problem\">\n<p><span class=\"os-number\">60<\/span><span class=\"os-divider\">.<\/span> [latex]\\sqrt[3]{24x^6}+\\sqrt[3]{81x^6}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339437571\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1167339437572\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339437571-solution\">61<\/a><span class=\"os-divider\">.<\/span> [latex]\\sqrt[4]{\\frac{162x^6}{16x^4}}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339437667\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1167339437668\" data-type=\"problem\">\n<p><span class=\"os-number\">62<\/span><span class=\"os-divider\">.<\/span> [latex]\\sqrt[3]{64y}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339437693\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1167339437694\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339437693-solution\">63<\/a><span class=\"os-divider\">.<\/span> [latex]\\sqrt[3]{128z^3}-\\sqrt[3]{-16z^3}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339437774\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1167339437775\" data-type=\"problem\">\n<p><span class=\"os-number\">64<\/span><span class=\"os-divider\">.<\/span> [latex]\\sqrt[5]{1,024c^{10}}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1167339437814\" data-depth=\"2\">\n<h3 data-type=\"title\">Real-World Applications<\/h3>\n<div id=\"fs-id1167339437820\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1167339437821\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339437820-solution\">65<\/a><span class=\"os-divider\">. <\/span>A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So the length of the guy wire can be found by evaluating [latex]\\sqrt{90,000+160,000}.[\/latex] What is the length of the guy wire?<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339437869\" class=\"material-set-2\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1167339437870\" data-type=\"problem\">\n<p><span class=\"os-number\">66<\/span><span class=\"os-divider\">. <\/span>A car accelerates at a rate of\u00a0[latex]6 - \\frac{\\sqrt{4}}{\\sqrt{t}} \\, \\text{m\/s}^2[\/latex] where [latex]t[\/latex] is the time in seconds after the car moves from rest. Simplify the expression.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1167339437932\" data-depth=\"2\">\n<h3 data-type=\"title\">Extensions<\/h3>\n<p id=\"fs-id1167339437937\">For the following exercises, simplify each expression.<\/p>\n<div id=\"fs-id1167339437940\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1167339437942\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339437940-solution\">67<\/a><span class=\"os-divider\">.<\/span> [latex]\\frac{\\sqrt{8}-\\sqrt{16}}{4-\\sqrt{2}}-2^{\\frac{1}{2}}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339438046\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1167339438047\" data-type=\"problem\">\n<p><span class=\"os-number\">68<\/span><span class=\"os-divider\">.<\/span> [latex]\\frac{4^{\\frac{3}{2}}-16^{\\frac{3}{2}}}{8^{\\frac{1}{3}}}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339438131\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1167339438132\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339438131-solution\">69<\/a><span class=\"os-divider\">.<\/span> [latex]\\frac{\\sqrt{mn^3}}{a^2\\sqrt{c^{-3}}} \\cdot \\frac{a^{-7}n^{-2}}{\\sqrt{m^2c^4}}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339438318\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1167339438319\" data-type=\"problem\">\n<p><span class=\"os-number\">70<\/span><span class=\"os-divider\">.<\/span> [latex]\\frac{a}{a-\\sqrt{c}}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339438349\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1167339438350\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339438349-solution\">71<\/a><span class=\"os-divider\">.<\/span> [latex]\\frac{x\\sqrt{64y} + 4\\sqrt{y}}{\\sqrt{128y}}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339438444\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1167339438445\" data-type=\"problem\">\n<p><span class=\"os-number\">72<\/span><span class=\"os-divider\">.<\/span> [latex]\\left(\\frac{\\sqrt{250x^2}}{\\sqrt{100b^3}}\\right)\\left(\\frac{7\\sqrt{b}}{\\sqrt{125x}}\\right)[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1167339438550\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1167339438552\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1167339438550-solution\">73<\/a><span class=\"os-divider\">.<\/span>[latex]\\sqrt{\\frac{\\sqrt[3]{64} + \\sqrt[4]{256}}{\\sqrt{64} + \\sqrt{256}}}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/section>\n<\/div>\n","protected":false},"author":158,"menu_order":3,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-86","chapter","type-chapter","status-publish","hentry"],"part":27,"_links":{"self":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/86","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":15,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/86\/revisions"}],"predecessor-version":[{"id":1800,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/86\/revisions\/1800"}],"part":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/parts\/27"}],"metadata":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/86\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/media?parent=86"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=86"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/contributor?post=86"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/license?post=86"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}