{"id":18324,"date":"2022-04-15T18:13:25","date_gmt":"2022-04-15T18:13:25","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/?post_type=chapter&#038;p=18324"},"modified":"2022-04-28T23:16:16","modified_gmt":"2022-04-28T23:16:16","slug":"cr-13-operations-with-square-roots","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/chapter\/cr-13-operations-with-square-roots\/","title":{"raw":"CR.13: Operations with Square Roots","rendered":"CR.13: Operations with 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 \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<\/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<div class=\"textbox exercises\">\r\n<h3>Example: Multiply radicals using the distributive property<\/h3>\r\nMultiply and simplify the expression.\r\n<p style=\"text-align: center;\">[latex](9-\\sqrt{7})(6+\\sqrt{7})[\/latex]<\/p>\r\n[reveal-answer q=\"134290\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"134290\"]\r\n\r\nFirst, use FOIL to multiply:\r\n[latex](9-\\sqrt{7})(6+\\sqrt{7})=9 \\cdot 6+9\\sqrt{7}-6\\sqrt{7}-(\\sqrt{7})^2[\/latex].\r\n\r\nNext, simplify each term, if possible:\r\n[latex]9 \\cdot 6+9\\sqrt{7}-6\\sqrt{7}-(\\sqrt{7})^2=54+9\\sqrt{7}-6\\sqrt{7}-7[\/latex]\r\n\r\nNow, simplify [latex]54-7[\/latex].\r\n[latex]54+9\\sqrt{7}-6\\sqrt{7}-7=47+9\\sqrt{7}-6\\sqrt{7}[\/latex]\r\n\r\nFinally, simplify [latex]9\\sqrt{7}-6\\sqrt{7}=3\\sqrt{7}[\/latex]:\r\n[latex]47+9\\sqrt{7}-6\\sqrt{7}=47+3\\sqrt{7}[\/latex]\r\nTherefore, [latex](9-\\sqrt{7})(6+\\sqrt{7})=47+3\\sqrt{7}[\/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\nMultiply and simplify [latex](1+\\sqrt{5})(4-\\sqrt{5})[\/latex].\r\n\r\n[reveal-answer q=\"481920\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"481920\"]\r\n\r\n[latex]3\\sqrt{5}-1[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=248375&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\r\n<h2>Adding and Subtracting Radical Expressions<\/h2>\r\nWe 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].\r\n<div class=\"textbox\">\r\n<h3>How To: Given a radical expression requiring addition or subtraction of square roots, solve.<\/h3>\r\n<ol>\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 class=\"textbox exercises\">\r\n<h3>Example: Adding Square Roots<\/h3>\r\nAdd [latex]5\\sqrt{12}+2\\sqrt{3}[\/latex].\r\n\r\n[reveal-answer q=\"742464\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"742464\"]\r\nWe 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\\left(2\\right)\\sqrt{3}[\/latex], which is [latex]10\\sqrt{3}[\/latex]. Now we can the terms have the same radicand so we can add.\r\n<p style=\"text-align: center;\">[latex]10\\sqrt{3}+2\\sqrt{3}=12\\sqrt{3}[\/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\nAdd [latex]\\sqrt{5}+6\\sqrt{20}[\/latex].\r\n\r\n[reveal-answer q=\"21382\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"21382\"]\r\n\r\n[latex]13\\sqrt{5}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2049&amp;theme=oea&amp;iframe_resize_id=mom1[\/embed]\r\n\r\n<\/div>\r\nWatch this video to see more examples of adding roots.\r\nhttps:\/\/youtu.be\/S3fGUeALy7E\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Subtracting Square Roots<\/h3>\r\nSubtract [latex]20\\sqrt{72{a}^{3}{b}^{4}c}-14\\sqrt{8{a}^{3}{b}^{4}c}[\/latex].\r\n\r\n[reveal-answer q=\"902648\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"902648\"]\r\n\r\nRewrite each term so they have equal radicands.\r\n<div style=\"text-align: center;\">\r\n<div style=\"text-align: center;\">[latex]\\begin{align} 20\\sqrt{72{a}^{3}{b}^{4}c}&amp; = 20\\sqrt{9}\\sqrt{4}\\sqrt{2}\\sqrt{a}\\sqrt{{a}^{2}}\\sqrt{{\\left({b}^{2}\\right)}^{2}}\\sqrt{c} \\\\ &amp; = 20\\left(3\\right)\\left(2\\right)a{b}^{2}\\sqrt{2ac} \\\\ &amp; = 120a{b}^{2}\\sqrt{2ac}\\\\ \\text{ } \\end{align}[\/latex]<\/div>\r\n<div><\/div>\r\n<div style=\"text-align: center;\">[latex]\\begin{align} 14\\sqrt{8{a}^{3}{b}^{4}c}&amp; = 14\\sqrt{2}\\sqrt{4}\\sqrt{a}\\sqrt{{a}^{2}}\\sqrt{{\\left({b}^{2}\\right)}^{2}}\\sqrt{c} \\\\ &amp; = 14\\left(2\\right)a{b}^{2}\\sqrt{2ac} \\\\ &amp; = 28a{b}^{2}\\sqrt{2ac} \\end{align}[\/latex]<\/div>\r\n<\/div>\r\nNow the terms have the same radicand so we can subtract.\r\n<div>[latex]120a{b}^{2}\\sqrt{2ac}-28a{b}^{2}\\sqrt{2ac}=92a{b}^{2}\\sqrt{2ac} \\\\ [\/latex]<\/div>\r\n<div>Note that we do not need an absolute value around the a because the [latex]a^3[\/latex] under the radical means that\u00a0<em>a<\/em> can't be negative.<\/div>\r\n<div>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nSubtract [latex]3\\sqrt{80x}-4\\sqrt{45x}[\/latex].\r\n\r\n[reveal-answer q=\"236912\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"236912\"]\r\n\r\n[latex]0[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=110419&amp;theme=oea&amp;iframe_resize_id=mom2[\/embed]\r\n\r\n<\/div>\r\nin the next video we show more examples of how to subtract radicals.\r\nhttps:\/\/youtu.be\/77TR9HsPZ6M\r\n<h2>Rationalize Denominators<\/h2>\r\n<div class=\"textbox examples\">\r\n<h3>Recall the identity property of Multiplication<\/h3>\r\nWe leverage an important and useful identity in this section in a technique commonly used in college algebra:\r\n<p style=\"text-align: center;\"><em> rewriting an expression by multiplying it by a well-chosen form of the number 1.<\/em><\/p>\r\nBecause the multiplicative identity states that\u00a0[latex]a\\cdot1=a[\/latex],\u00a0we are able to multiply the top and bottom of any fraction by the same number without changing its value. We use this idea when we\u00a0<em>rationalize the denominator.<\/em>\r\n\r\n<\/div>\r\nWhen 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>rationalizing the denominator<\/em>.\r\n\r\nWe 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.\r\n\r\nFor 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]\\dfrac{\\sqrt{c}}{\\sqrt{c}}[\/latex].\r\n\r\nFor 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].\r\n<div class=\"textbox\">\r\n<h3>How To: Given an expression with a single square root radical term in the denominator, rationalize the denominator.<\/h3>\r\n<ol>\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 class=\"textbox exercises\">\r\n<h3>Example: Rationalizing a Denominator Containing a Single Term<\/h3>\r\nWrite [latex]\\dfrac{2\\sqrt{3}}{3\\sqrt{10}}[\/latex] in simplest form.\r\n\r\n[reveal-answer q=\"982148\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"982148\"]\r\n\r\nThe radical in the denominator is [latex]\\sqrt{10}[\/latex]. So multiply the fraction by [latex]\\dfrac{\\sqrt{10}}{\\sqrt{10}}[\/latex]. Then simplify.\r\n<div style=\"text-align: center;\">[latex]\\begin{align}\\frac{2\\sqrt{3}}{3\\sqrt{10}}\\cdot \\frac{\\sqrt{10}}{\\sqrt{10}} &amp;= \\frac{2\\sqrt{30}}{30} \\\\ &amp;= \\frac{\\sqrt{30}}{15}\\end{align}[\/latex]<\/div>\r\n<div>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nWrite [latex]\\dfrac{12\\sqrt{3}}{\\sqrt{2}}[\/latex] in simplest form.\r\n\r\n[reveal-answer q=\"497322\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"497322\"]\r\n\r\n[latex]6\\sqrt{6}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2765&amp;theme=oea&amp;iframe_resize_id=mom4[\/embed]\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given an expression with a radical term and a constant in the denominator, rationalize the denominator.<\/h3>\r\n<ol>\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 class=\"textbox exercises\">\r\n<h3>Example: Rationalizing a Denominator Containing Two Terms<\/h3>\r\nWrite [latex]\\dfrac{4}{1+\\sqrt{5}}[\/latex] in simplest form.\r\n\r\n[reveal-answer q=\"726340\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"726340\"]\r\n\r\nBegin 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]\\dfrac{1-\\sqrt{5}}{1-\\sqrt{5}}[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{align}\\frac{4}{1+\\sqrt{5}}\\cdot \\frac{1-\\sqrt{5}}{1-\\sqrt{5}} &amp;= \\frac{4 - 4\\sqrt{5}}{-4} &amp;&amp; \\text{Use the distributive property}. \\\\ &amp;=\\sqrt{5}-1 &amp;&amp; \\text{Simplify}. \\end{align}[\/latex]<\/div>\r\n<div>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nWrite [latex]\\dfrac{7}{2+\\sqrt{3}}[\/latex] in simplest form.\r\n\r\n[reveal-answer q=\"132932\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"132932\"]\r\n\r\n[latex]14 - 7\\sqrt{3}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=3441&amp;theme=oea&amp;iframe_resize_id=mom5[\/embed]\r\n\r\n<\/div>\r\nhttps:\/\/www.youtube.com\/watch?v=vINRIRgeKqU&amp;feature=youtu.be\r\n\r\n&nbsp;","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<li>Use the quotient rule to simplify square roots.<\/li>\n<li>Add and subtract square roots.<\/li>\n<li>Rationalize denominators.<\/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<div class=\"textbox exercises\">\n<h3>Example: Multiply radicals using the distributive property<\/h3>\n<p>Multiply and simplify the expression.<\/p>\n<p style=\"text-align: center;\">[latex](9-\\sqrt{7})(6+\\sqrt{7})[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q134290\">Show Solution<\/span><\/p>\n<div id=\"q134290\" class=\"hidden-answer\" style=\"display: none\">\n<p>First, use FOIL to multiply:<br \/>\n[latex](9-\\sqrt{7})(6+\\sqrt{7})=9 \\cdot 6+9\\sqrt{7}-6\\sqrt{7}-(\\sqrt{7})^2[\/latex].<\/p>\n<p>Next, simplify each term, if possible:<br \/>\n[latex]9 \\cdot 6+9\\sqrt{7}-6\\sqrt{7}-(\\sqrt{7})^2=54+9\\sqrt{7}-6\\sqrt{7}-7[\/latex]<\/p>\n<p>Now, simplify [latex]54-7[\/latex].<br \/>\n[latex]54+9\\sqrt{7}-6\\sqrt{7}-7=47+9\\sqrt{7}-6\\sqrt{7}[\/latex]<\/p>\n<p>Finally, simplify [latex]9\\sqrt{7}-6\\sqrt{7}=3\\sqrt{7}[\/latex]:<br \/>\n[latex]47+9\\sqrt{7}-6\\sqrt{7}=47+3\\sqrt{7}[\/latex]<br \/>\nTherefore, [latex](9-\\sqrt{7})(6+\\sqrt{7})=47+3\\sqrt{7}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Multiply and simplify [latex](1+\\sqrt{5})(4-\\sqrt{5})[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q481920\">Show Solution<\/span><\/p>\n<div id=\"q481920\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]3\\sqrt{5}-1[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm248375\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=248375&#38;theme=oea&#38;iframe_resize_id=ohm248375&#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<h2>Adding and Subtracting Radical Expressions<\/h2>\n<p>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\">\n<h3>How To: Given a radical expression requiring addition or subtraction of square roots, solve.<\/h3>\n<ol>\n<li>Simplify each radical expression.<\/li>\n<li>Add or subtract expressions with equal radicands.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Adding Square Roots<\/h3>\n<p>Add [latex]5\\sqrt{12}+2\\sqrt{3}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q742464\">Show Solution<\/span><\/p>\n<div id=\"q742464\" class=\"hidden-answer\" style=\"display: none\">\nWe 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\\left(2\\right)\\sqrt{3}[\/latex], which is [latex]10\\sqrt{3}[\/latex]. Now we can 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<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Add [latex]\\sqrt{5}+6\\sqrt{20}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q21382\">Show Solution<\/span><\/p>\n<div id=\"q21382\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]13\\sqrt{5}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm2049\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2049&#38;theme=oea&#38;iframe_resize_id=ohm2049&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Watch this video to see more examples of adding roots.<br \/>\n<iframe loading=\"lazy\" id=\"oembed-2\" title=\"Adding Radicals That Requires Simplifying\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/S3fGUeALy7E?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Subtracting Square Roots<\/h3>\n<p>Subtract [latex]20\\sqrt{72{a}^{3}{b}^{4}c}-14\\sqrt{8{a}^{3}{b}^{4}c}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q902648\">Show Solution<\/span><\/p>\n<div id=\"q902648\" class=\"hidden-answer\" style=\"display: none\">\n<p>Rewrite each term so they have equal radicands.<\/p>\n<div style=\"text-align: center;\">\n<div style=\"text-align: center;\">[latex]\\begin{align} 20\\sqrt{72{a}^{3}{b}^{4}c}& = 20\\sqrt{9}\\sqrt{4}\\sqrt{2}\\sqrt{a}\\sqrt{{a}^{2}}\\sqrt{{\\left({b}^{2}\\right)}^{2}}\\sqrt{c} \\\\ & = 20\\left(3\\right)\\left(2\\right)a{b}^{2}\\sqrt{2ac} \\\\ & = 120a{b}^{2}\\sqrt{2ac}\\\\ \\text{ } \\end{align}[\/latex]<\/div>\n<div><\/div>\n<div style=\"text-align: center;\">[latex]\\begin{align} 14\\sqrt{8{a}^{3}{b}^{4}c}& = 14\\sqrt{2}\\sqrt{4}\\sqrt{a}\\sqrt{{a}^{2}}\\sqrt{{\\left({b}^{2}\\right)}^{2}}\\sqrt{c} \\\\ & = 14\\left(2\\right)a{b}^{2}\\sqrt{2ac} \\\\ & = 28a{b}^{2}\\sqrt{2ac} \\end{align}[\/latex]<\/div>\n<\/div>\n<p>Now the terms have the same radicand so we can subtract.<\/p>\n<div>[latex]120a{b}^{2}\\sqrt{2ac}-28a{b}^{2}\\sqrt{2ac}=92a{b}^{2}\\sqrt{2ac} \\\\[\/latex]<\/div>\n<div>Note that we do not need an absolute value around the a because the [latex]a^3[\/latex] under the radical means that\u00a0<em>a<\/em> can&#8217;t be negative.<\/div>\n<div><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Subtract [latex]3\\sqrt{80x}-4\\sqrt{45x}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q236912\">Show Solution<\/span><\/p>\n<div id=\"q236912\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]0[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm110419\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=110419&#38;theme=oea&#38;iframe_resize_id=ohm110419&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>in the next video we show more examples of how to subtract radicals.<br \/>\n<iframe loading=\"lazy\" id=\"oembed-3\" title=\"Subtracting Radicals (Basic With No Simplifying)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/77TR9HsPZ6M?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Rationalize Denominators<\/h2>\n<div class=\"textbox examples\">\n<h3>Recall the identity property of Multiplication<\/h3>\n<p>We leverage an important and useful identity in this section in a technique commonly used in college algebra:<\/p>\n<p style=\"text-align: center;\"><em> rewriting an expression by multiplying it by a well-chosen form of the number 1.<\/em><\/p>\n<p>Because the multiplicative identity states that\u00a0[latex]a\\cdot1=a[\/latex],\u00a0we are able to multiply the top and bottom of any fraction by the same number without changing its value. We use this idea when we\u00a0<em>rationalize the denominator.<\/em><\/p>\n<\/div>\n<p>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>rationalizing the denominator<\/em>.<\/p>\n<p>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>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]\\dfrac{\\sqrt{c}}{\\sqrt{c}}[\/latex].<\/p>\n<p>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\">\n<h3>How To: Given an expression with a single square root radical term in the denominator, rationalize the denominator.<\/h3>\n<ol>\n<li>Multiply the numerator and denominator by the radical in the denominator.<\/li>\n<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Rationalizing a Denominator Containing a Single Term<\/h3>\n<p>Write [latex]\\dfrac{2\\sqrt{3}}{3\\sqrt{10}}[\/latex] in simplest form.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q982148\">Show Solution<\/span><\/p>\n<div id=\"q982148\" class=\"hidden-answer\" style=\"display: none\">\n<p>The radical in the denominator is [latex]\\sqrt{10}[\/latex]. So multiply the fraction by [latex]\\dfrac{\\sqrt{10}}{\\sqrt{10}}[\/latex]. Then simplify.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align}\\frac{2\\sqrt{3}}{3\\sqrt{10}}\\cdot \\frac{\\sqrt{10}}{\\sqrt{10}} &= \\frac{2\\sqrt{30}}{30} \\\\ &= \\frac{\\sqrt{30}}{15}\\end{align}[\/latex]<\/div>\n<div><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Write [latex]\\dfrac{12\\sqrt{3}}{\\sqrt{2}}[\/latex] in simplest form.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q497322\">Show Solution<\/span><\/p>\n<div id=\"q497322\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]6\\sqrt{6}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm2765\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2765&#38;theme=oea&#38;iframe_resize_id=ohm2765&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given an expression with a radical term and a constant in the denominator, rationalize the denominator.<\/h3>\n<ol>\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 class=\"textbox exercises\">\n<h3>Example: Rationalizing a Denominator Containing Two Terms<\/h3>\n<p>Write [latex]\\dfrac{4}{1+\\sqrt{5}}[\/latex] in simplest form.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q726340\">Show Solution<\/span><\/p>\n<div id=\"q726340\" class=\"hidden-answer\" style=\"display: none\">\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]\\dfrac{1-\\sqrt{5}}{1-\\sqrt{5}}[\/latex].<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align}\\frac{4}{1+\\sqrt{5}}\\cdot \\frac{1-\\sqrt{5}}{1-\\sqrt{5}} &= \\frac{4 - 4\\sqrt{5}}{-4} && \\text{Use the distributive property}. \\\\ &=\\sqrt{5}-1 && \\text{Simplify}. \\end{align}[\/latex]<\/div>\n<div><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Write [latex]\\dfrac{7}{2+\\sqrt{3}}[\/latex] in simplest form.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q132932\">Show Solution<\/span><\/p>\n<div id=\"q132932\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]14 - 7\\sqrt{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm3441\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=3441&#38;theme=oea&#38;iframe_resize_id=ohm3441&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" title=\"Ex:  Rationalize the Denominator of a Radical Expression - Conjugate\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/vINRIRgeKqU?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n","protected":false},"author":264444,"menu_order":13,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-18324","chapter","type-chapter","status-publish","hentry"],"part":18142,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/18324","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/users\/264444"}],"version-history":[{"count":11,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/18324\/revisions"}],"predecessor-version":[{"id":18708,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/18324\/revisions\/18708"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/parts\/18142"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/18324\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/media?parent=18324"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapter-type?post=18324"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/contributor?post=18324"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/license?post=18324"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}