{"id":49,"date":"2023-06-21T13:22:28","date_gmt":"2023-06-21T13:22:28","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/evaluate-and-simplify-square-roots\/"},"modified":"2023-07-04T04:54:53","modified_gmt":"2023-07-04T04:54:53","slug":"evaluate-and-simplify-square-roots","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/evaluate-and-simplify-square-roots\/","title":{"raw":"\u25aa   Evaluate and Simplify Square Roots","rendered":"\u25aa   Evaluate and Simplify Square Roots"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Evaluate principal square roots.<\/li>\r\n \t<li>Use the product rule to simplify square roots.<\/li>\r\n<\/ul>\r\n<\/div>\r\nWhen 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.\r\n\r\nIn 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>principal square root<\/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.\r\n\r\nThe principal square root of [latex]a[\/latex] is written as [latex]\\sqrt{a}[\/latex]. The symbol is called a <strong>radical<\/strong>, the term under the symbol is called the <strong>radicand<\/strong>, and the entire expression is called a <strong>radical expression<\/strong>.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/24203630\/CNX_CAT_Figure_01_03_002.jpg\" 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.\" \/>\r\n<div class=\"textbox\">\r\n<h3>A General Note: Principal Square Root<\/h3>\r\nThe <strong>principal square root<\/strong> of [latex]a[\/latex] is the nonnegative number that, when multiplied by itself, equals [latex]a[\/latex]. It is written as a <strong>radical expression<\/strong>, with a symbol called a <strong>radical<\/strong> over the term called the <strong>radicand<\/strong>: [latex]\\sqrt{a}[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>Q &amp; A<\/h3>\r\n<h3><strong>Does<\/strong> [latex]\\sqrt{25}=\\pm 5[\/latex]?<\/h3>\r\n<em>No. Although both<\/em> [latex]{5}^{2}[\/latex] <em>and<\/em> [latex]{\\left(-5\\right)}^{2}[\/latex] <em>are<\/em> [latex]25[\/latex], <em>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 class=\"textbox exercises\">\r\n<h3>Example: Evaluating Square Roots<\/h3>\r\nEvaluate each expression.\r\n<ol>\r\n \t<li>[latex]\\sqrt{100}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{\\sqrt{16}}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{25+144}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{49}-\\sqrt{81}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"849035\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"849035\"]\r\n<ol>\r\n \t<li>[latex]\\sqrt{100}=10[\/latex] because [latex]{10}^{2}=100[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{\\sqrt{16}}=\\sqrt{4}=2[\/latex] because [latex]{4}^{2}=16[\/latex] and [latex]{2}^{2}=4[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{25+144}=\\sqrt{169}=13[\/latex] because [latex]{13}^{2}=169[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{49}-\\sqrt{81}=7 - 9=-2[\/latex] because [latex]{7}^{2}=49[\/latex] and [latex]{9}^{2}=81[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>Q &amp; A<\/h3>\r\n<h3>For [latex]\\sqrt{25+144}[\/latex], can we find the square roots before adding?<\/h3>\r\n<em>No.<\/em> [latex]\\sqrt{25}+\\sqrt{144}=5+12=17[\/latex]. <em>This is not equivalent to<\/em> [latex]\\sqrt{25+144}=13[\/latex]. <em>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 class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nEvaluate each expression.\r\n<ol>\r\n \t<li>[latex]\\sqrt{225}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{\\sqrt{81}}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{25 - 9}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{36}+\\sqrt{121}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"98241\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"98241\"]\r\n<ol>\r\n \t<li>[latex]15[\/latex]<\/li>\r\n \t<li>[latex]3[\/latex]<\/li>\r\n \t<li>[latex]4[\/latex]<\/li>\r\n \t<li>[latex]17[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=14119&amp;theme=oea&amp;iframe_resize_id=mom1[\/embed]\r\n\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=109776&amp;theme=oea&amp;iframe_resize_id=mom10[\/embed]\r\n\r\n<\/div>\r\n<h2>Use the Product Rule to Simplify Square Roots<\/h2>\r\nTo 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>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.\r\n<div class=\"textbox\">\r\n<h3>A General Note: The Product Rule for Simplifying Square Roots<\/h3>\r\nIf [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].\r\n<div style=\"text-align: center;\">[latex]\\sqrt{ab}=\\sqrt{a}\\cdot \\sqrt{b}[\/latex]<\/div>\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given a square root radical expression, use the product rule to simplify it.<strong>\r\n<\/strong><\/h3>\r\n<ol>\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 class=\"textbox examples\">\r\n<h3>Recall Prime Factorization<\/h3>\r\nIt can be helpful, when simplifying square roots, to write the radicand as a product of primes in order to find perfect squares under the radical.\r\n\r\nExample. [latex]\\sqrt{288} \\quad=\\quad \\sqrt{2\\cdot3^2\\cdot4^2} \\quad=\\quad 3\\cdot4\\cdot \\sqrt{2} \\quad=\\quad 12\\sqrt{2}.[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Using the Product Rule to Simplify Square Roots<\/h3>\r\nSimplify the radical expression.\r\n<ol>\r\n \t<li>[latex]\\sqrt{300}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{162{a}^{5}{b}^{4}}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"483887\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"483887\"]\r\n\r\n1.\r\n[latex]\\begin{align}&amp;\\sqrt{100\\cdot 3} &amp;&amp; \\text{Factor perfect square from radicand}. \\\\ &amp;\\sqrt{100}\\cdot \\sqrt{3} &amp;&amp; \\text{Write radical expression as product of radical expressions}. \\\\ &amp;10\\sqrt{3} &amp;&amp; \\text{Simplify}. \\\\ \\text{ }\\end{align}[\/latex]\r\n\r\n2.\r\n[latex]\\begin{align}&amp;\\sqrt{81{a}^{4}{b}^{4}\\cdot 2a} &amp;&amp; \\text{Factor perfect square from radicand}. \\\\ &amp;\\sqrt{81{a}^{4}{b}^{4}}\\cdot \\sqrt{2a} &amp;&amp; \\text{Write radical expression as product of radical expressions}. \\\\ &amp;9{a}^{2}{b}^{2}\\sqrt{2a} &amp;&amp; \\text{Simplify}. \\end{align}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nSimplify [latex]\\sqrt{50{x}^{2}{y}^{3}z}[\/latex].\r\n\r\n[reveal-answer q=\"86548\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"86548\"]\r\n\r\n[latex]5|x||y|\\sqrt{2yz}[\/latex]. Notice the absolute value signs around <em>x<\/em> and <em>y<\/em>? That\u2019s because their value must be positive!\r\n\r\n[\/hidden-answer]\r\n\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=110285&amp;theme=oea&amp;iframe_resize_id=mom20[\/embed]\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given the product of multiple radical expressions, use the product rule to combine them into one radical expression.<strong>\r\n<\/strong><\/h3>\r\n<ol>\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 class=\"textbox exercises\">\r\n<h3>Example: Using the Product Rule to Simplify the Product of Multiple Square Roots<\/h3>\r\nSimplify the radical expression.\r\n<p style=\"text-align: center;\">[latex]\\sqrt{12}\\cdot \\sqrt{3}[\/latex]<\/p>\r\n[reveal-answer q=\"134287\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"134287\"]\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;\\sqrt{12\\cdot 3} &amp;&amp; \\text{Express the product as a single radical expression}. \\\\ &amp;\\sqrt{36} &amp;&amp; \\text{Simplify}. \\\\ &amp;6 \\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nSimplify [latex]\\sqrt{50x}\\cdot \\sqrt{2x}[\/latex] assuming [latex]x&gt;0[\/latex].\r\n\r\n[reveal-answer q=\"481919\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"481919\"]\r\n\r\n[latex]10x[\/latex]\r\nBecause [latex]x&gt;0[\/latex], we do not need an absolute values.\r\n\r\n[\/hidden-answer]\r\n\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=110272&amp;theme=oea&amp;iframe_resize_id=mom10[\/embed]\r\n\r\n<\/div>\r\n<h2>Using the Quotient Rule to Simplify Square Roots<\/h2>\r\nJust 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>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{\\dfrac{5}{2}}[\/latex] as [latex]\\dfrac{\\sqrt{5}}{\\sqrt{2}}[\/latex].\r\n<div class=\"textbox\">\r\n<h3>A General Note: The Quotient Rule for Simplifying Square Roots<\/h3>\r\nThe square root of the quotient [latex]\\dfrac{a}{b}[\/latex] is equal to the quotient of the square roots of [latex]a[\/latex] and [latex]b[\/latex], where [latex]b\\ne 0[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\sqrt{\\dfrac{a}{b}}=\\dfrac{\\sqrt{a}}{\\sqrt{b}}[\/latex]<\/div>\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given a radical expression, use the quotient rule to simplify it.<strong>\r\n<\/strong><\/h3>\r\n<ol>\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 class=\"textbox examples\">\r\n<h3>Recall Simplifying Fractions<\/h3>\r\n<p style=\"text-align: left;\">To simplify fractions, find common factors in the numerator and denominator that cancel.<\/p>\r\n<p style=\"text-align: left;\">Example:\u00a0 \u00a0 \u00a0 [latex]\\dfrac{24}{32}\\quad=\\quad\\dfrac{\\cancel{2}\\cdot\\cancel{2}\\cdot\\cancel{2}\\cdot3}{\\cancel{2}\\cdot\\cancel{2}\\cdot\\cancel{2}\\cdot2\\cdot2}\\quad=\\quad\\dfrac{3}{2\\cdot2}\\quad=\\quad\\dfrac{3}{4}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Using the Quotient Rule to Simplify Square Roots<\/h3>\r\nSimplify the radical expression.\r\n<p style=\"text-align: center;\">[latex]\\sqrt{\\dfrac{5}{36}}[\/latex]<\/p>\r\n[reveal-answer q=\"317945\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"317945\"]\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;\\frac{\\sqrt{5}}{\\sqrt{36}} &amp;&amp; \\text{Write as quotient of two radical expressions}. \\\\ &amp;\\frac{\\sqrt{5}}{6} &amp;&amp; \\text{Simplify denominator}. \\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nSimplify [latex]\\sqrt{\\dfrac{2{x}^{2}}{9{y}^{4}}}[\/latex].\r\n\r\n[reveal-answer q=\"671876\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"671876\"]\r\n\r\n[latex]\\dfrac{x\\sqrt{2}}{3{y}^{2}}[\/latex]. We do not need the absolute value signs for [latex]{y}^{2}[\/latex] because that term will always be nonnegative.\r\n\r\n[\/hidden-answer]\r\n\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=110287&amp;theme=oea&amp;iframe_resize_id=mom20[\/embed]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Using the Quotient Rule to Simplify an Expression with Two Square Roots<\/h3>\r\nSimplify the radical expression.\r\n<p style=\"text-align: center;\">[latex]\\dfrac{\\sqrt{234{x}^{11}y}}{\\sqrt{26{x}^{7}y}}[\/latex]<\/p>\r\n[reveal-answer q=\"520119\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"520119\"]\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;\\sqrt{\\frac{234{x}^{11}y}{26{x}^{7}y}} &amp;&amp; \\text{Combine numerator and denominator into one radical expression}. \\\\ &amp;\\sqrt{9{x}^{4}} &amp;&amp; \\text{Simplify fraction}. \\\\ &amp;3{x}^{2} &amp;&amp; \\text{Simplify square root}. \\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nSimplify [latex]\\dfrac{\\sqrt{9{a}^{5}{b}^{14}}}{\\sqrt{3{a}^{4}{b}^{5}}}[\/latex].\r\n\r\n[reveal-answer q=\"157179\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"157179\"]\r\n\r\n[latex]{b}^{4}\\sqrt{3ab}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=110387&amp;theme=oea&amp;iframe_resize_id=mom10[\/embed]\r\n\r\n<\/div>\r\nIn the following video you will see more examples of how to simplify radical expressions with variables.\r\nhttps:\/\/youtu.be\/q7LqsKPoAKo","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Evaluate principal square roots.<\/li>\n<li>Use the product rule to simplify square roots.<\/li>\n<\/ul>\n<\/div>\n<p>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>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>principal square root<\/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>The principal square root of [latex]a[\/latex] is written as [latex]\\sqrt{a}[\/latex]. The symbol is called a <strong>radical<\/strong>, the term under the symbol is called the <strong>radicand<\/strong>, and the entire expression is called a <strong>radical expression<\/strong>.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/24203630\/CNX_CAT_Figure_01_03_002.jpg\" 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.\" \/><\/p>\n<div class=\"textbox\">\n<h3>A General Note: Principal Square Root<\/h3>\n<p>The <strong>principal square root<\/strong> of [latex]a[\/latex] is the nonnegative number that, when multiplied by itself, equals [latex]a[\/latex]. It is written as a <strong>radical expression<\/strong>, with a symbol called a <strong>radical<\/strong> over the term called the <strong>radicand<\/strong>: [latex]\\sqrt{a}[\/latex].<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<h3><strong>Does<\/strong> [latex]\\sqrt{25}=\\pm 5[\/latex]?<\/h3>\n<p><em>No. Although both<\/em> [latex]{5}^{2}[\/latex] <em>and<\/em> [latex]{\\left(-5\\right)}^{2}[\/latex] <em>are<\/em> [latex]25[\/latex], <em>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 class=\"textbox exercises\">\n<h3>Example: Evaluating Square Roots<\/h3>\n<p>Evaluate each expression.<\/p>\n<ol>\n<li>[latex]\\sqrt{100}[\/latex]<\/li>\n<li>[latex]\\sqrt{\\sqrt{16}}[\/latex]<\/li>\n<li>[latex]\\sqrt{25+144}[\/latex]<\/li>\n<li>[latex]\\sqrt{49}-\\sqrt{81}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q849035\">Show Solution<\/span><\/p>\n<div id=\"q849035\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]\\sqrt{100}=10[\/latex] because [latex]{10}^{2}=100[\/latex]<\/li>\n<li>[latex]\\sqrt{\\sqrt{16}}=\\sqrt{4}=2[\/latex] because [latex]{4}^{2}=16[\/latex] and [latex]{2}^{2}=4[\/latex]<\/li>\n<li>[latex]\\sqrt{25+144}=\\sqrt{169}=13[\/latex] because [latex]{13}^{2}=169[\/latex]<\/li>\n<li>[latex]\\sqrt{49}-\\sqrt{81}=7 - 9=-2[\/latex] because [latex]{7}^{2}=49[\/latex] and [latex]{9}^{2}=81[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<h3>For [latex]\\sqrt{25+144}[\/latex], can we find the square roots before adding?<\/h3>\n<p><em>No.<\/em> [latex]\\sqrt{25}+\\sqrt{144}=5+12=17[\/latex]. <em>This is not equivalent to<\/em> [latex]\\sqrt{25+144}=13[\/latex]. <em>The order of operations requires us to add the terms in the radicand before finding the square root.<\/em><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Evaluate each expression.<\/p>\n<ol>\n<li>[latex]\\sqrt{225}[\/latex]<\/li>\n<li>[latex]\\sqrt{\\sqrt{81}}[\/latex]<\/li>\n<li>[latex]\\sqrt{25 - 9}[\/latex]<\/li>\n<li>[latex]\\sqrt{36}+\\sqrt{121}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q98241\">Show Solution<\/span><\/p>\n<div id=\"q98241\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]15[\/latex]<\/li>\n<li>[latex]3[\/latex]<\/li>\n<li>[latex]4[\/latex]<\/li>\n<li>[latex]17[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm14119\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=14119&#38;theme=oea&#38;iframe_resize_id=ohm14119&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm109776\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=109776&#38;theme=oea&#38;iframe_resize_id=ohm109776&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Use the Product Rule to Simplify Square Roots<\/h2>\n<p>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>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\">\n<h3>A General Note: The Product Rule for Simplifying Square Roots<\/h3>\n<p>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<div style=\"text-align: center;\">[latex]\\sqrt{ab}=\\sqrt{a}\\cdot \\sqrt{b}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a square root radical expression, use the product rule to simplify it.<strong><br \/>\n<\/strong><\/h3>\n<ol>\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 class=\"textbox examples\">\n<h3>Recall Prime Factorization<\/h3>\n<p>It can be helpful, when simplifying square roots, to write the radicand as a product of primes in order to find perfect squares under the radical.<\/p>\n<p>Example. [latex]\\sqrt{288} \\quad=\\quad \\sqrt{2\\cdot3^2\\cdot4^2} \\quad=\\quad 3\\cdot4\\cdot \\sqrt{2} \\quad=\\quad 12\\sqrt{2}.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using the Product Rule to Simplify Square Roots<\/h3>\n<p>Simplify the radical expression.<\/p>\n<ol>\n<li>[latex]\\sqrt{300}[\/latex]<\/li>\n<li>[latex]\\sqrt{162{a}^{5}{b}^{4}}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q483887\">Show Solution<\/span><\/p>\n<div id=\"q483887\" class=\"hidden-answer\" style=\"display: none\">\n<p>1.<br \/>\n[latex]\\begin{align}&\\sqrt{100\\cdot 3} && \\text{Factor perfect square from radicand}. \\\\ &\\sqrt{100}\\cdot \\sqrt{3} && \\text{Write radical expression as product of radical expressions}. \\\\ &10\\sqrt{3} && \\text{Simplify}. \\\\ \\text{ }\\end{align}[\/latex]<\/p>\n<p>2.<br \/>\n[latex]\\begin{align}&\\sqrt{81{a}^{4}{b}^{4}\\cdot 2a} && \\text{Factor perfect square from radicand}. \\\\ &\\sqrt{81{a}^{4}{b}^{4}}\\cdot \\sqrt{2a} && \\text{Write radical expression as product of radical expressions}. \\\\ &9{a}^{2}{b}^{2}\\sqrt{2a} && \\text{Simplify}. \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Simplify [latex]\\sqrt{50{x}^{2}{y}^{3}z}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q86548\">Show Solution<\/span><\/p>\n<div id=\"q86548\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]5|x||y|\\sqrt{2yz}[\/latex]. Notice the absolute value signs around <em>x<\/em> and <em>y<\/em>? That\u2019s because their value must be positive!<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm110285\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=110285&#38;theme=oea&#38;iframe_resize_id=ohm110285&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given the product of multiple radical expressions, use the product rule to combine them into one radical expression.<strong><br \/>\n<\/strong><\/h3>\n<ol>\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 class=\"textbox exercises\">\n<h3>Example: Using the Product Rule to Simplify the Product of Multiple Square Roots<\/h3>\n<p>Simplify the radical expression.<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt{12}\\cdot \\sqrt{3}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q134287\">Show Solution<\/span><\/p>\n<div id=\"q134287\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\">[latex]\\begin{align}&\\sqrt{12\\cdot 3} && \\text{Express the product as a single radical expression}. \\\\ &\\sqrt{36} && \\text{Simplify}. \\\\ &6 \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Simplify [latex]\\sqrt{50x}\\cdot \\sqrt{2x}[\/latex] assuming [latex]x>0[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q481919\">Show Solution<\/span><\/p>\n<div id=\"q481919\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]10x[\/latex]<br \/>\nBecause [latex]x>0[\/latex], we do not need an absolute values.<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm110272\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=110272&#38;theme=oea&#38;iframe_resize_id=ohm110272&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Using the Quotient Rule to Simplify Square Roots<\/h2>\n<p>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>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{\\dfrac{5}{2}}[\/latex] as [latex]\\dfrac{\\sqrt{5}}{\\sqrt{2}}[\/latex].<\/p>\n<div class=\"textbox\">\n<h3>A General Note: The Quotient Rule for Simplifying Square Roots<\/h3>\n<p>The square root of the quotient [latex]\\dfrac{a}{b}[\/latex] is equal to the quotient of the square roots of [latex]a[\/latex] and [latex]b[\/latex], where [latex]b\\ne 0[\/latex].<\/p>\n<div style=\"text-align: center;\">[latex]\\sqrt{\\dfrac{a}{b}}=\\dfrac{\\sqrt{a}}{\\sqrt{b}}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a radical expression, use the quotient rule to simplify it.<strong><br \/>\n<\/strong><\/h3>\n<ol>\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 class=\"textbox examples\">\n<h3>Recall Simplifying Fractions<\/h3>\n<p style=\"text-align: left;\">To simplify fractions, find common factors in the numerator and denominator that cancel.<\/p>\n<p style=\"text-align: left;\">Example:\u00a0 \u00a0 \u00a0 [latex]\\dfrac{24}{32}\\quad=\\quad\\dfrac{\\cancel{2}\\cdot\\cancel{2}\\cdot\\cancel{2}\\cdot3}{\\cancel{2}\\cdot\\cancel{2}\\cdot\\cancel{2}\\cdot2\\cdot2}\\quad=\\quad\\dfrac{3}{2\\cdot2}\\quad=\\quad\\dfrac{3}{4}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using the Quotient Rule to Simplify Square Roots<\/h3>\n<p>Simplify the radical expression.<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt{\\dfrac{5}{36}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q317945\">Show Solution<\/span><\/p>\n<div id=\"q317945\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\">[latex]\\begin{align}&\\frac{\\sqrt{5}}{\\sqrt{36}} && \\text{Write as quotient of two radical expressions}. \\\\ &\\frac{\\sqrt{5}}{6} && \\text{Simplify denominator}. \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Simplify [latex]\\sqrt{\\dfrac{2{x}^{2}}{9{y}^{4}}}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q671876\">Show Solution<\/span><\/p>\n<div id=\"q671876\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\dfrac{x\\sqrt{2}}{3{y}^{2}}[\/latex]. We do not need the absolute value signs for [latex]{y}^{2}[\/latex] because that term will always be nonnegative.<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm110287\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=110287&#38;theme=oea&#38;iframe_resize_id=ohm110287&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using the Quotient Rule to Simplify an Expression with Two Square Roots<\/h3>\n<p>Simplify the radical expression.<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{\\sqrt{234{x}^{11}y}}{\\sqrt{26{x}^{7}y}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q520119\">Show Solution<\/span><\/p>\n<div id=\"q520119\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\">[latex]\\begin{align}&\\sqrt{\\frac{234{x}^{11}y}{26{x}^{7}y}} && \\text{Combine numerator and denominator into one radical expression}. \\\\ &\\sqrt{9{x}^{4}} && \\text{Simplify fraction}. \\\\ &3{x}^{2} && \\text{Simplify square root}. \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Simplify [latex]\\dfrac{\\sqrt{9{a}^{5}{b}^{14}}}{\\sqrt{3{a}^{4}{b}^{5}}}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q157179\">Show Solution<\/span><\/p>\n<div id=\"q157179\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]{b}^{4}\\sqrt{3ab}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm110387\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=110387&#38;theme=oea&#38;iframe_resize_id=ohm110387&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the following video you will see more examples of how to simplify radical expressions with variables.<br \/>\n<iframe loading=\"lazy\" id=\"oembed-1\" title=\"Simplify Square Roots with Variables\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/q7LqsKPoAKo?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-49\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><li>Simplify Square Roots With Variables. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/q7LqsKPoAKo\">https:\/\/youtu.be\/q7LqsKPoAKo<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Question ID 14119. <strong>Authored by<\/strong>: James Sousa. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License, CC-BY + GPL<\/li><li>Question ID 109776, 110285, 110272, 110287, 110387. <strong>Authored by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License, CC-BY + GPL<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: OpenStax College Algebra. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":395986,"menu_order":18,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"College Algebra\",\"author\":\"OpenStax College Algebra\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and 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