{"id":18235,"date":"2022-04-08T05:30:00","date_gmt":"2022-04-08T05:30:00","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/?post_type=chapter&#038;p=18235"},"modified":"2022-04-28T23:15:20","modified_gmt":"2022-04-28T23:15:20","slug":"cr-7-radicals","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/chapter\/cr-7-radicals\/","title":{"raw":"CR.7: Radicals","rendered":"CR.7: Radicals"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Simplify expressions with square roots using the order of operations<\/li>\r\n \t<li>Simplify expressions with square roots that contain variables<\/li>\r\n \t<li>Add and subtract square roots<\/li>\r\n \t<li>Simplify Nth roots<\/li>\r\n \t<li>Write radicals as rational exponents<\/li>\r\n \t<li>Multiply and divide radical expressions<\/li>\r\n \t<li>Use the product raised to a power rule to multiply radical expressions<\/li>\r\n<\/ul>\r\n<\/div>\r\n&nbsp;\r\n<h3 data-type=\"title\">Square Roots and the Order of Operations<\/h3>\r\nWhen using the order of operations to simplify an expression that has square roots, we treat the radical sign as a grouping symbol. We simplify any expressions under the radical sign before performing other operations.\r\n\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: \u24d0 [latex]\\sqrt{25}+\\sqrt{144}[\/latex] \u24d1 [latex]\\sqrt{25+144}[\/latex].\r\n\r\nSolution\r\n<table id=\"eip-id1168466048150\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td>\u24d0 Use the order of operations.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\sqrt{25}+\\sqrt{144}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify each radical.<\/td>\r\n<td>[latex]5+12[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add.<\/td>\r\n<td>[latex]17[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168469785439\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td>\u24d1 Use the order of operations.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\sqrt{25+144}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add under the radical sign.<\/td>\r\n<td>[latex]\\sqrt{169}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]13[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146635[\/ohm_question]\r\n\r\n[ohm_question]146636[\/ohm_question]\r\n\r\n<\/div>\r\nNotice the different answers in parts \u24d0 and \u24d1 of the example above. It is important to follow the order of operations correctly. In \u24d0, we took each square root first and then added them. In \u24d1, we added under the radical sign first and then found the square root.\r\n<h2 data-type=\"title\">Simplify Variable Expressions with Square Roots<\/h2>\r\nExpressions with square root that we have looked at so far have not had any variables. What happens when we have to find a square root of a variable expression?\r\nConsider [latex]\\sqrt{9{x}^{2}}[\/latex], where [latex]x\\ge 0[\/latex]. Can you think of an expression whose square is [latex]9{x}^{2}?[\/latex]\r\n\r\n[latex]\\begin{array}{ccc}\\hfill {\\left(?\\right)}^{2}&amp; =&amp; 9{x}^{2}\\hfill \\\\ \\hfill {\\left(3x\\right)}^{2}&amp; =&amp; 9{x}^{2}\\text{so}\\sqrt{9{x}^{2}}=3x\\hfill \\end{array}[\/latex]\r\nWhen we use a variable in a square root expression, for our work, we will assume that the variable represents a non-negative number. In every example and exercise that follows, each variable in a square root expression is greater than or equal to zero.\r\n\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]\\sqrt{{x}^{2}}[\/latex], where [latex]x\\ge 0[\/latex]\r\n[reveal-answer q=\"394349\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"394349\"]\r\n\r\nSolution\r\nThink about what we would have to square to get [latex]{x}^{2}[\/latex] . Algebraically, [latex]{\\left(?\\right)}^{2}={x}^{2}[\/latex]\r\n<table id=\"eip-id1168467419284\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\sqrt{{x}^{2}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Since [latex]{\\left(x\\right)}^{2}={x}^{2}[\/latex]<\/td>\r\n<td>[latex]x[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146637[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]\\sqrt{16{x}^{2}}[\/latex].\r\n[reveal-answer q=\"924940\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"924940\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168466092245\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\sqrt{16{x}^{2}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\text{Since}{\\left(4x\\right)}^{2}=16{x}^{2}[\/latex]<\/td>\r\n<td>[latex]4x[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146638[\/ohm_question]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]-\\sqrt{81{y}^{2}}[\/latex].\r\n[reveal-answer q=\"614317\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"614317\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168468389692\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-\\sqrt{81{y}^{2}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\text{Since}{\\left(9y\\right)}^{2}=81{y}^{2}[\/latex]<\/td>\r\n<td>[latex]-9y[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146639[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]\\sqrt{36{x}^{2}{y}^{2}}[\/latex].\r\n[reveal-answer q=\"164861\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"164861\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168466004603\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\sqrt{36{x}^{2}{y}^{2}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\text{Since}{\\left(6xy\\right)}^{2}=36{x}^{2}{y}^{2}[\/latex]<\/td>\r\n<td>[latex]6xy[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146640[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<h2>Operations on Square Roots<\/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>Nth Roots and Rational Exponents<\/h2>\r\n<h3>Using Rational Roots<\/h3>\r\nAlthough 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.\r\nSuppose 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.\r\n\r\nThe <em>n<\/em>th root of [latex]a[\/latex] is a number that, when raised to the <em>n<\/em>th power, gives [latex]a[\/latex]. For example, [latex]-3[\/latex] is the 5th root of [latex]-243[\/latex] because [latex]{\\left(-3\\right)}^{5}=-243[\/latex]. If [latex]a[\/latex] is a real number with at least one <em>n<\/em>th root, then the <strong>principal <em>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>n<\/em>th power, equals [latex]a[\/latex].\r\n\r\nThe principal <em>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.\r\n<div class=\"textbox\">\r\n<h3>A General Note: Principal <em>n<\/em>th Root<\/h3>\r\nIf [latex]a[\/latex] is a real number with at least one <em>n<\/em>th root, then the <strong>principal <em>n<\/em>th root<\/strong> 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>n<\/em>th power, equals [latex]a[\/latex]. The <strong>index<\/strong> of the radical is [latex]n[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Simplifying <em>n<\/em>th Roots<\/h3>\r\nSimplify each of the following:\r\n<ol>\r\n \t<li>[latex]\\sqrt[5]{-32}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt[4]{4}\\cdot \\sqrt[4]{1,024}[\/latex]<\/li>\r\n \t<li>[latex]-\\sqrt[3]{\\dfrac{8{x}^{6}}{125}}[\/latex]<\/li>\r\n \t<li>[latex]8\\sqrt[4]{3}-\\sqrt[4]{48}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"149528\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"149528\"]\r\n<ol>\r\n \t<li>[latex]\\sqrt[5]{-32}=-2[\/latex] because [latex]{\\left(-2\\right)}^{5}=-32 \\\\ \\text{ }[\/latex]<\/li>\r\n \t<li>First, express the product as a single radical expression. [latex]\\sqrt[4]{4\\text{,}096}=8[\/latex] because [latex]{8}^{4}=4,096[\/latex]<\/li>\r\n \t<li>[latex]\\begin{align}\\\\ &amp;\\frac{-\\sqrt[3]{8{x}^{6}}}{\\sqrt[3]{125}} &amp;&amp; \\text{Write as quotient of two radical expressions}. \\\\ &amp;\\frac{-2{x}^{2}}{5} &amp;&amp; \\text{Simplify}. \\\\ \\end{align}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{align}\\\\ &amp;8\\sqrt[4]{3}-2\\sqrt[4]{3} &amp;&amp; \\text{Simplify to get equal radicands}. \\\\ &amp;6\\sqrt[4]{3} &amp;&amp; \\text{Add}. \\end{align}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nSimplify.\r\n<ol>\r\n \t<li>[latex]\\sqrt[3]{-216}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{3\\sqrt[4]{80}}{\\sqrt[4]{5}}[\/latex]<\/li>\r\n \t<li>[latex]6\\sqrt[3]{9,000}+7\\sqrt[3]{576}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"15987\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"15987\"]\r\n<ol>\r\n \t<li>[latex]-6[\/latex]<\/li>\r\n \t<li>[latex]6[\/latex]<\/li>\r\n \t<li>[latex]88\\sqrt[3]{9}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2564&amp;theme=oea&amp;iframe_resize_id=mom1[\/embed]\r\n\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2565&amp;theme=oea&amp;iframe_resize_id=mom2[\/embed]\r\n\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2567&amp;theme=oea&amp;iframe_resize_id=mom3[\/embed]\r\n\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2592&amp;theme=oea&amp;iframe_resize_id=mom4[\/embed]\r\n\r\n<\/div>\r\n<h2>Simplify Radical Expressions<\/h2>\r\n<strong>Radical expressions<\/strong> are expressions that contain radicals. Radical expressions come in many forms, from simple and familiar, such as[latex] \\sqrt{16}[\/latex], to quite complicated, as in [latex] \\sqrt[3]{250{{x}^{4}}y}[\/latex].\r\n\r\nTo simplify complicated radical expressions, we can use some definitions and rules from simplifying exponents. Recall the <strong>Product Raised to a Power Rule<\/strong> from when you studied exponents. This rule states that the product of two or more non-zero numbers raised to a power is equal to the product of each number raised to the same power. In math terms, it is written [latex]\\left(ab\\right)^{x}=a^{x}\\cdot{b}^{x}[\/latex]. So, for example, you can use the rule to rewrite [latex] {{\\left( 3x \\right)}^{2}}[\/latex] as [latex] {{3}^{2}}\\cdot {{x}^{2}}=9\\cdot {{x}^{2}}=9{{x}^{2}}[\/latex].\r\n\r\nNow instead of using the exponent [latex]2[\/latex], use the exponent [latex] \\frac{1}{2}[\/latex]. The exponent is distributed in the same way.\r\n<p style=\"text-align: center;\">[latex] {{\\left( 3x \\right)}^{\\frac{1}{2}}}={{3}^{\\frac{1}{2}}}\\cdot {{x}^{\\frac{1}{2}}}[\/latex]<\/p>\r\nAnd since you know that raising a number to the [latex] \\frac{1}{2}[\/latex] power is the same as taking the square root of that number, you can also write it this way.\r\n<p style=\"text-align: center;\">[latex] \\sqrt{3x}=\\sqrt{3}\\cdot \\sqrt{x}[\/latex]<\/p>\r\nLook at that\u2014you can think of any number underneath a radical as the <i>product of separate factors<\/i>, each underneath its own radical.\r\n<div class=\"textbox shaded\">\r\n<h3>A Product Raised to a Power Rule or sometimes called The Square Root of a Product Rule<\/h3>\r\nFor any real numbers <i>a<\/i> and <i>b<\/i>, [latex] \\sqrt{ab}=\\sqrt{a}\\cdot \\sqrt{b}[\/latex].\r\n\r\nFor example: [latex] \\sqrt{100}=\\sqrt{10}\\cdot \\sqrt{10}[\/latex], and [latex] \\sqrt{75}=\\sqrt{25}\\cdot \\sqrt{3}[\/latex]\r\n\r\n<\/div>\r\nThis rule is important because it helps you think of one radical as the product of multiple radicals. If you can identify perfect squares within a radical, as with [latex] \\sqrt{(2\\cdot 2)(2\\cdot 2)(3\\cdot 3})[\/latex], you can rewrite the expression as the product of multiple perfect squares: [latex] \\sqrt{{{2}^{2}}}\\cdot \\sqrt{{{2}^{2}}}\\cdot \\sqrt{{{3}^{2}}}[\/latex].\r\n<p class=\"p1\">The square root of a product rule will help us simplify roots that are not perfect as is shown the following example.<\/p>\r\n\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\sqrt{63}[\/latex]\r\n<div class=\"qa-wrapper\" style=\"display: block;\">\r\n\r\n<span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q908978\">Show Solution<\/span>\r\n<div id=\"q908978\" class=\"hidden-answer\" style=\"display: none;\">\r\n\r\n[latex]63[\/latex] is not a perfect square so we can use the square root of a product rule to simplify any factors that are perfect squares.\r\nFactor [latex]63[\/latex] into [latex]7[\/latex] and [latex]9[\/latex].\r\n[latex] \\sqrt{7\\cdot 9}[\/latex]\r\n[latex]9[\/latex] is a perfect square, [latex]9=3^2[\/latex], therefore we can rewrite the radicand.\r\n\r\n[latex] \\sqrt{7\\cdot {{3}^{2}}}[\/latex]\r\n\r\nUsing the Product Raised to a Power rule, separate the radical into the product of two factors, each under a radical.\r\n[latex] \\sqrt{7}\\cdot \\sqrt{{{3}^{2}}}[\/latex]\r\nTake the square root of [latex]3^{2}[\/latex].\r\n[latex] \\sqrt{7}\\cdot 3[\/latex]\r\nRearrange factors so the integer appears before the radical and then multiply. This is done so that it is clear that only the [latex]7[\/latex] is under the radical, not the [latex]3[\/latex].\r\n[latex] 3\\cdot \\sqrt{7}[\/latex]\r\nThe answer is [latex]3\\sqrt{7}[\/latex].\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nThe final answer [latex] 3\\sqrt{7}[\/latex] may look a bit odd, but it is in simplified form. You can read this as \u201cthree radical seven\u201d or \u201cthree times the square root of seven.\u201d\r\n\r\nThe following video shows more examples of how to simplify square roots that do not have perfect square radicands.\r\n\r\n<iframe src=\"https:\/\/www.youtube.com\/embed\/oRd7aBCsmfU?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n\r\nBefore we move on to simplifying more complex radicals with variables, we need to learn about an important behavior of square roots with variables in the radicand.\r\n\r\nConsider the expression [latex] \\sqrt{{{x}^{2}}}[\/latex]. This looks like it should be equal to <i>x<\/i>, right? Test some values for <i>x<\/i> and see what happens.\r\n\r\nIn the chart below, look along each row and determine whether the value of <i>x<\/i> is the same as the value of [latex] \\sqrt{{{x}^{2}}}[\/latex]. Where are they equal? Where are they not equal?\r\n\r\nAfter doing that for each row, look again and determine whether the value of [latex] \\sqrt{{{x}^{2}}}[\/latex] is the same as the value of [latex]\\left|x\\right|[\/latex].\r\n<table style=\"width: 40%;\">\r\n<thead>\r\n<tr style=\"height: 30px;\">\r\n<th style=\"height: 30px;\">[latex]x[\/latex]<\/th>\r\n<th style=\"height: 30px;\">[latex]x^{2}[\/latex]<\/th>\r\n<th style=\"height: 30px;\">[latex]\\sqrt{x^{2}}[\/latex]<\/th>\r\n<th style=\"height: 30px;\">[latex]\\left|x\\right|[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"height: 30px;\">[latex]\u22125[\/latex]<\/td>\r\n<td style=\"height: 30px;\">[latex]25[\/latex]<\/td>\r\n<td style=\"height: 30px;\">[latex]5[\/latex]<\/td>\r\n<td style=\"height: 30px;\">[latex]5[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"height: 30px;\">[latex]\u22122[\/latex]<\/td>\r\n<td style=\"height: 30px;\">[latex]4[\/latex]<\/td>\r\n<td style=\"height: 30px;\">[latex]2[\/latex]<\/td>\r\n<td style=\"height: 30px;\">[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\">[latex]0[\/latex]<\/td>\r\n<td style=\"height: 15px;\">[latex]0[\/latex]<\/td>\r\n<td style=\"height: 15px;\">[latex]0[\/latex]<\/td>\r\n<td style=\"height: 15px;\">[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\">[latex]6[\/latex]<\/td>\r\n<td style=\"height: 15px;\">[latex]36[\/latex]<\/td>\r\n<td style=\"height: 15px;\">[latex]6[\/latex]<\/td>\r\n<td style=\"height: 15px;\">[latex]6[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15.125px;\">\r\n<td style=\"height: 15.125px;\">[latex]10[\/latex]<\/td>\r\n<td style=\"height: 15.125px;\">[latex]100[\/latex]<\/td>\r\n<td style=\"height: 15.125px;\">[latex]10[\/latex]<\/td>\r\n<td style=\"height: 15.125px;\">[latex]10[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNotice\u2014in cases where <i>x<\/i> is a negative number, [latex]\\sqrt{x^{2}}\\neq{x}[\/latex]! However, in all cases [latex]\\sqrt{x^{2}}=\\left|x\\right|[\/latex]. You need to consider this fact when simplifying radicals with an even index that contain variables, because by definition [latex]\\sqrt{x^{2}}[\/latex] is always nonnegative.\r\n<div class=\"textbox shaded\">\r\n<h3>Taking the Square Root of a Radical Expression<\/h3>\r\nWhen finding the square root of an expression that contains variables raised to an even power, remember that [latex]\\sqrt{x^{2}}=\\left|x\\right|[\/latex].\r\n\r\nExamples: [latex]\\sqrt{9x^{2}}=3\\left|x\\right|[\/latex], and [latex]\\sqrt{16{{x}^{2}}{{y}^{2}}}=4\\left|xy\\right|[\/latex]\r\n\r\n<\/div>\r\nWe will combine this with the square root of a product rule in our next example to simplify an expression with three variables in the radicand.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\sqrt{{{a}^{3}}{{b}^{5}}{{c}^{2}}}[\/latex]\r\n<div class=\"qa-wrapper\" style=\"display: block;\">\r\n\r\n<span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q141094\">Show Solution<\/span>\r\n<div id=\"q141094\" class=\"hidden-answer\" style=\"display: none;\">\r\n\r\nFactor to find variables with even exponents.\r\n\r\n[latex] \\sqrt{{{a}^{2}}\\cdot a\\cdot {{b}^{4}}\\cdot b\\cdot {{c}^{2}}}[\/latex]\r\n\r\nRewrite [latex]b^{4}[\/latex] as [latex]\\left(b^{2}\\right)^{2}[\/latex].\r\n\r\n[latex] \\sqrt{{{a}^{2}}\\cdot a\\cdot {{({{b}^{2}})}^{2}}\\cdot b\\cdot {{c}^{2}}}[\/latex]\r\n\r\nSeparate the squared factors into individual radicals.\r\n\r\n[latex] \\sqrt{{{a}^{2}}}\\cdot \\sqrt{{{({{b}^{2}})}^{2}}}\\cdot \\sqrt{{{c}^{2}}}\\cdot \\sqrt{a\\cdot b}[\/latex]\r\n\r\nTake the square root of each radical. Remember that [latex] \\sqrt{{{a}^{2}}}=\\left| a \\right|[\/latex].\r\n\r\n[latex] \\left| a \\right|\\cdot {{b}^{2}}\\cdot \\left|{c}\\right|\\cdot \\sqrt{a\\cdot b}[\/latex]\r\n\r\nSimplify and multiply.\r\n\r\n[latex] \\left| ac \\right|{{b}^{2}}\\sqrt{ab}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<h3>Analysis of the Solution<\/h3>\r\nWhy did we not write [latex]b^2[\/latex] as [latex]|b^2|[\/latex]? Because when you square a number, you will always get a positive result, so the principal square root of [latex]\\left(b^2\\right)^2[\/latex] will always be non-negative. One tip for knowing when to apply the absolute value after simplifying any even indexed root is to look at the final exponent on your variable terms. If the exponent is odd \u2013 including [latex]1[\/latex] \u2013 add an absolute value. This applies to simplifying any root with an even index, as we will see in later examples.\r\n\r\nIn the following video, you will see more examples of how to simplify radical expressions with variables.\r\n\r\n<iframe src=\"https:\/\/www.youtube.com\/embed\/q7LqsKPoAKo?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n\r\nWe will show another example where the simplified expression contains variables with both odd and even powers.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\sqrt{9{{x}^{6}}{{y}^{4}}}[\/latex]\r\n<div class=\"qa-wrapper\" style=\"display: block;\">\r\n\r\n<span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q41297\">Show Solution<\/span>\r\n<div id=\"q41297\" class=\"hidden-answer\" style=\"display: none;\">\r\n\r\nFactor to find identical pairs.\r\n<p style=\"text-align: center;\">[latex] \\sqrt{3\\cdot 3\\cdot {{x}^{3}}\\cdot {{x}^{3}}\\cdot {{y}^{2}}\\cdot {{y}^{2}}}[\/latex]<\/p>\r\nRewrite the pairs as perfect squares.\r\n<p style=\"text-align: center;\">[latex] \\sqrt{{{3}^{2}}\\cdot {{\\left( {{x}^{3}} \\right)}^{2}}\\cdot {{\\left( {{y}^{2}} \\right)}^{2}}}[\/latex]<\/p>\r\nSeparate into individual radicals.\r\n<p style=\"text-align: center;\">[latex] \\sqrt{{{3}^{2}}}\\cdot \\sqrt{{{\\left( {{x}^{3}} \\right)}^{2}}}\\cdot \\sqrt{{{\\left( {{y}^{2}} \\right)}^{2}}}[\/latex]<\/p>\r\nSimplify.\r\n<p style=\"text-align: center;\">[latex] 3{{x}^{3}}{{y}^{2}}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">Because x has an odd power, we will add the absolute value for our final solution.<\/p>\r\n<p style=\"text-align: center;\">[latex] 3|{{x}^{3}}|{{y}^{2}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nIn our next example, we will start with an expression written with a rational exponent. You will see that you can use a similar process \u2013 factoring and sorting terms into squares \u2013 to simplify this expression.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] {{(36{{x}^{4}})}^{\\frac{1}{2}}}[\/latex]\r\n<div class=\"qa-wrapper\" style=\"display: block;\">\r\n\r\n<span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q554375\">Show Solution<\/span>\r\n<div id=\"q554375\" class=\"hidden-answer\" style=\"display: none;\">\r\n\r\nRewrite the expression with the fractional exponent as a radical.\r\n<p style=\"text-align: center;\">[latex] \\sqrt{36{{x}^{4}}}[\/latex]<\/p>\r\nFind the square root of both the coefficient and the variable.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r} \\sqrt{{{6}^{2}}\\cdot {{x}^{4}}}\\\\\\sqrt{{{6}^{2}}}\\cdot \\sqrt{{{x}^{4}}}\\\\\\sqrt{{{6}^{2}}}\\cdot \\sqrt{{{({{x}^{2}})}^{2}}}\\\\6\\cdot{x}^{2}\\end{array}[\/latex]<\/p>\r\nThe answer is [latex]6{{x}^{2}}[\/latex].\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nHere is one more example with perfect squares.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\sqrt{49{{x}^{10}}{{y}^{8}}}[\/latex]\r\n<div class=\"qa-wrapper\" style=\"display: block;\">\r\n\r\n<span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q283065\">Show Solution<\/span>\r\n<div id=\"q283065\" class=\"hidden-answer\" style=\"display: none;\">\r\n\r\nLook for squared numbers and variables. Factor [latex]49[\/latex] into [latex]7\\cdot7[\/latex], [latex]x^{10}[\/latex] into [latex]x^{5}\\cdot{x}^{5}[\/latex], and [latex]y^{8}[\/latex] into [latex]y^{4}\\cdot{y}^{4}[\/latex].\r\n<p style=\"text-align: center;\">[latex] \\sqrt{7\\cdot 7\\cdot {{x}^{5}}\\cdot {{x}^{5}}\\cdot {{y}^{4}}\\cdot {{y}^{4}}}[\/latex]<\/p>\r\nRewrite the pairs as squares.\r\n<p style=\"text-align: center;\">[latex] \\sqrt{{{7}^{2}}\\cdot {{({{x}^{5}})}^{2}}\\cdot {{({{y}^{4}})}^{2}}}[\/latex]<\/p>\r\nSeparate the squared factors into individual radicals.\r\n<p style=\"text-align: center;\">[latex] \\sqrt{{{7}^{2}}}\\cdot \\sqrt{{{({{x}^{5}})}^{2}}}\\cdot \\sqrt{{{({{y}^{4}})}^{2}}}[\/latex]<\/p>\r\nTake the square root of each radical using the rule that [latex] \\sqrt{{{x}^{2}}}=x[\/latex].\r\n<p style=\"text-align: center;\">[latex] 7\\cdot {{x}^{5}}\\cdot {{y}^{4}}[\/latex]<\/p>\r\nMultiply.\r\n<p style=\"text-align: center;\">[latex] 7{{x}^{5}}{{y}^{4}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<h2>Simplify Cube Roots<\/h2>\r\nWe can use the same techniques we have used for simplifying square roots to simplify higher order roots. For example, to simplify a cube root, the goal is to find factors under the radical that are perfect cubes so that you can take their cube root. We no longer need to be concerned about whether we have identified the principal root since we are now finding cube roots. Focus on finding identical trios of factors as you simplify.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\sqrt[3]{40{{m}^{5}}}[\/latex]\r\n<div class=\"qa-wrapper\" style=\"display: block;\">\r\n\r\n<span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q617053\">Show Solution<\/span>\r\n<div id=\"q617053\" class=\"hidden-answer\" style=\"display: none;\">\r\n\r\nFactor [latex]40[\/latex] into prime factors.\r\n\r\n[latex] \\sqrt[3]{5\\cdot 2\\cdot 2\\cdot 2\\cdot {{m}^{5}}}[\/latex]\r\n\r\nSince you are looking for the cube root, you need to find factors that appear [latex]3[\/latex] times under the radical. Rewrite [latex] 2\\cdot 2\\cdot 2[\/latex] as [latex] {{2}^{3}}[\/latex].\r\n\r\n[latex] \\sqrt[3]{{{2}^{3}}\\cdot 5\\cdot {{m}^{5}}}[\/latex]\r\n\r\nRewrite [latex] {{m}^{5}}[\/latex] as [latex] {{m}^{3}}\\cdot {{m}^{2}}[\/latex].\r\n\r\n[latex] \\sqrt[3]{{{2}^{3}}\\cdot 5\\cdot {{m}^{3}}\\cdot {{m}^{2}}}[\/latex]\r\n\r\nRewrite the expression as a product of multiple radicals.\r\n\r\n[latex] \\sqrt[3]{{{2}^{3}}}\\cdot \\sqrt[3]{5}\\cdot \\sqrt[3]{{{m}^{3}}}\\cdot \\sqrt[3]{{{m}^{2}}}[\/latex]\r\n\r\nSimplify and multiply.\r\n\r\n[latex] 2\\cdot \\sqrt[3]{5}\\cdot m\\cdot \\sqrt[3]{{{m}^{2}}}[\/latex]\r\n\r\nThe answer is [latex]2m\\sqrt[3]{5{{m}^{2}}}[\/latex].\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nRemember that you can take the cube root of a negative expression. In the next example, we will simplify a cube root with a negative radicand.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\sqrt[3]{-27{{x}^{4}}{{y}^{3}}}[\/latex]\r\n<div class=\"qa-wrapper\" style=\"display: block;\">\r\n\r\n<span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q670300\">Show Solution<\/span>\r\n<div id=\"q670300\" class=\"hidden-answer\" style=\"display: none;\">\r\n\r\nFactor the expression into cubes.\r\n\r\nSeparate the cubed factors into individual radicals.\r\n\r\n[latex]\\begin{array}{r}\\sqrt[3]{-1\\cdot 27\\cdot {{x}^{4}}\\cdot {{y}^{3}}}\\\\\\sqrt[3]{{{(-1)}^{3}}\\cdot {{(3)}^{3}}\\cdot {{x}^{3}}\\cdot x\\cdot {{y}^{3}}}\\\\\\sqrt[3]{{{(-1)}^{3}}}\\cdot \\sqrt[3]{{{(3)}^{3}}}\\cdot \\sqrt[3]{{{x}^{3}}}\\cdot \\sqrt[3]{x}\\cdot \\sqrt[3]{{{y}^{3}}}\\end{array}[\/latex]\r\n\r\nSimplify the cube roots.\r\n\r\n[latex] -1\\cdot 3\\cdot x\\cdot y\\cdot \\sqrt[3]{x}[\/latex]\r\n\r\nThe answer is [latex] \\sqrt[3]{-27{{x}^{4}}{{y}^{3}}}=-3xy\\sqrt[3]{x}[\/latex].\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nYou could check your answer by performing the inverse operation. If you are right, when you cube [latex] -3xy\\sqrt[3]{x}[\/latex] you should get [latex] -27{{x}^{4}}{{y}^{3}}[\/latex].\r\n\r\n[latex] \\begin{array}{l}\\left( -3xy\\sqrt[3]{x} \\right)\\left( -3xy\\sqrt[3]{x} \\right)\\left( -3xy\\sqrt[3]{x} \\right)\\\\-3\\cdot -3\\cdot -3\\cdot x\\cdot x\\cdot x\\cdot y\\cdot y\\cdot y\\cdot \\sqrt[3]{x}\\cdot \\sqrt[3]{x}\\cdot \\sqrt[3]{x}\\\\-27\\cdot {{x}^{3}}\\cdot {{y}^{3}}\\cdot \\sqrt[3]{{{x}^{3}}}\\\\-27{{x}^{3}}{{y}^{3}}\\cdot x\\\\-27{{x}^{4}}{{y}^{3}}\\end{array}[\/latex]\r\n\r\nYou can also skip the step of factoring out the negative one once you are comfortable with identifying cubes.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\sqrt[3]{-24{{a}^{5}}}[\/latex]\r\n<div class=\"qa-wrapper\" style=\"display: block;\">\r\n\r\n<span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q473861\">Show Solution<\/span>\r\n<div id=\"q473861\" class=\"hidden-answer\" style=\"display: none;\">\r\n\r\nFactor [latex]\u221224[\/latex] to find perfect cubes. Here, [latex]\u22121[\/latex] and [latex]8[\/latex] are the perfect cubes.\r\n<p style=\"text-align: center;\">[latex] \\sqrt[3]{-1\\cdot 8\\cdot 3\\cdot {{a}^{5}}}[\/latex]<\/p>\r\nFactor variables. You are looking for cube exponents, so you factor [latex]a^{5}[\/latex] into [latex]a^{3}[\/latex] and [latex]a^{2}[\/latex].\r\n<p style=\"text-align: center;\">[latex] \\sqrt[3]{{{(-1)}^{3}}\\cdot {{2}^{3}}\\cdot 3\\cdot {{a}^{3}}\\cdot {{a}^{2}}}[\/latex]<\/p>\r\nSeparate the factors into individual radicals.\r\n<p style=\"text-align: center;\">[latex] \\sqrt[3]{{{(-1)}^{3}}}\\cdot \\sqrt[3]{{{2}^{3}}}\\cdot \\sqrt[3]{{{a}^{3}}}\\cdot \\sqrt[3]{3\\cdot {{a}^{2}}}[\/latex]<\/p>\r\nSimplify, using the property [latex] \\sqrt[3]{{{x}^{3}}}=x[\/latex].\r\n<p style=\"text-align: center;\">[latex] -1\\cdot 2\\cdot a\\cdot \\sqrt[3]{3\\cdot {{a}^{2}}}[\/latex]<\/p>\r\nThis is the simplest form of this expression; all cubes have been pulled out of the radical expression.\r\n<p style=\"text-align: center;\">[latex] -2a\\sqrt[3]{3{{a}^{2}}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nIn the following video, we show more examples of simplifying cube roots.\r\n\r\n<iframe src=\"https:\/\/www.youtube.com\/embed\/BtJruOpmHCE?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n<h2>Simplifying Fourth Roots<\/h2>\r\nNow let us move to simplifying fourth degree roots. No matter what root you are simplifying, the same idea applies: find cubes for cube roots, powers of four for fourth roots, etc. Recall that when your simplified expression contains an even indexed radical and a variable factor with an odd exponent, you need to apply an absolute value.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\sqrt[4]{81{{x}^{8}}{{y}^{3}}}[\/latex]\r\n<div class=\"qa-wrapper\" style=\"display: block;\">\r\n\r\n<span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q295348\">Show Solution<\/span>\r\n<div id=\"q295348\" class=\"hidden-answer\" style=\"display: none;\">\r\n\r\nRewrite the expression.\r\n\r\n[latex] \\sqrt[4]{81}\\cdot \\sqrt[4]{{{x}^{8}}}\\cdot \\sqrt[4]{{{y}^{3}}}[\/latex]\r\n\r\nFactor each radicand.\r\n\r\n[latex] \\sqrt[4]{3\\cdot 3\\cdot 3\\cdot 3}\\cdot \\sqrt[4]{{{x}^{2}}\\cdot {{x}^{2}}\\cdot {{x}^{2}}\\cdot {{x}^{2}}}\\cdot \\sqrt[4]{{{y}^{3}}}[\/latex]\r\n\r\nSimplify.\r\n\r\n[latex]\\begin{array}{r}\\sqrt[4]{{{3}^{4}}}\\cdot \\sqrt[4]{{{({{x}^{2}})}^{4}}}\\cdot \\sqrt[4]{{{y}^{3}}}\\\\3\\cdot {{x}^{2}}\\cdot \\sqrt[4]{{{y}^{3}}}\\end{array}[\/latex]\r\n\r\nThe answer is [latex]\\sqrt[4]{81x^{8}y^{3}}=3x^{2}\\sqrt[4]{y^{3}} [\/latex].\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nAn alternative method to factoring is to rewrite the expression with rational exponents, then use the rules of exponents to simplify. You may find that you prefer one method over the other. Either way, it is nice to have options. We will show the last example again, using this idea.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\sqrt[4]{81{{x}^{8}}{{y}^{3}}}[\/latex]\r\n<div class=\"qa-wrapper\" style=\"display: block;\">\r\n\r\n<span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q324337\">Show Solution<\/span>\r\n<div id=\"q324337\" class=\"hidden-answer\" style=\"display: none;\">\r\n\r\nRewrite the radical using rational exponents.\r\n\r\n[latex] {{(81{{x}^{8}}{{y}^{3}})}^{\\frac{1}{4}}}[\/latex]\r\n\r\nUse the rules of exponents to simplify the expression.\r\n\r\n[latex] \\begin{array}{r}{{81}^{\\frac{1}{4}}}\\cdot {{x}^{\\frac{8}{4}}}\\cdot {{y}^{\\frac{3}{4}}}\\\\{{(3\\cdot 3\\cdot 3\\cdot 3)}^{\\frac{1}{4}}}{{x}^{2}}{{y}^{\\frac{3}{4}}}\\\\{{({{3}^{4}})}^{\\frac{1}{4}}}{{x}^{2}}{{y}^{\\frac{3}{4}}}\\\\3{{x}^{2}}{{y}^{\\frac{3}{4}}}\\end{array}[\/latex]\r\n\r\nChange the expression with the rational exponent back to radical form.\r\n\r\n[latex] 3{{x}^{2}}\\sqrt[4]{{{y}^{3}}}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nIn the following video, we show another example of how to simplify a fourth and fifth root.\r\n\r\n<iframe src=\"https:\/\/www.youtube.com\/embed\/op2LEb0YRyw?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n\r\nFor our last example, we will simplify a more complicated expression, [latex]\\dfrac{10{{b}^{2}}{{c}^{2}}}{c\\sqrt[3]{8{{b}^{4}}}}[\/latex]<i>.<\/i><i> <\/i>This expression has two variables, a fraction, and a radical. Let us take it step-by-step and see if using fractional exponents can help us simplify it.\r\nWe will start by simplifying the denominator since this is where the radical sign is located. Recall that an exponent in the denominator of a fraction can be rewritten as a negative exponent.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex]\\dfrac{10{{b}^{2}}{{c}^{2}}}{c\\sqrt[3]{8{{b}^{4}}}}[\/latex]\r\n<div class=\"qa-wrapper\" style=\"display: block;\">\r\n\r\n<span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q962386\">Show Solution<\/span>\r\n<div id=\"q962386\" class=\"hidden-answer\" style=\"display: none;\">\r\n\r\nSeparate the factors in the denominator.\r\n\r\n[latex] \\frac{10{{b}^{2}}{{c}^{2}}}{c\\cdot \\sqrt[3]{8}\\cdot \\sqrt[3]{{{b}^{4}}}}[\/latex]\r\n\r\nTake the cube root of [latex]8[\/latex], which is [latex]2[\/latex].\r\n\r\n[latex] \\frac{10{{b}^{2}}{{c}^{2}}}{c\\cdot 2\\cdot \\sqrt[3]{{{b}^{4}}}}[\/latex]\r\n\r\nRewrite the radical using a fractional exponent.\r\n\r\n[latex] \\frac{10{{b}^{2}}{{c}^{2}}}{c\\cdot 2\\cdot {{b}^{\\frac{4}{3}}}}[\/latex]\r\n\r\nRewrite the fraction as a series of factors in order to cancel factors (see next step).\r\n\r\n[latex] \\frac{10}{2}\\cdot \\frac{{{c}^{2}}}{c}\\cdot \\frac{{{b}^{2}}}{{{b}^{\\frac{4}{3}}}}[\/latex]\r\n\r\nSimplify the constant and <i>c<\/i> factors.\r\n\r\n[latex] 5\\cdot c\\cdot \\frac{{{b}^{2}}}{{{b}^{\\frac{4}{3}}}}[\/latex]\r\n\r\nUse the rule of negative exponents, <i>n<\/i><sup>\u2013<\/sup><i><sup>x<\/sup><\/i><i>=<\/i>[latex] \\frac{1}{{{n}^{x}}}[\/latex], to rewrite [latex] \\frac{1}{{{b}^{\\tfrac{4}{3}}}}[\/latex] as [latex] {{b}^{-\\tfrac{4}{3}}}[\/latex].\r\n\r\n[latex] 5c{{b}^{2}}{{b}^{-\\ \\frac{4}{3}}}[\/latex]\r\n\r\nCombine the <i>b<\/i> factors by adding the exponents.\r\n\r\n[latex] 5c{{b}^{\\frac{2}{3}}}[\/latex]\r\n\r\nChange the expression with the fractional exponent back to radical form. By convention, an expression is not usually considered simplified if it has a fractional exponent or a radical in the denominator.\r\n\r\n[latex] 5c\\sqrt[3]{{{b}^{2}}}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nWell, that took a while, but you did it. You applied what you know about fractional exponents, negative exponents, and the rules of exponents to simplify the expression.\r\n\r\nIn our last video, we show how to use rational exponents to simplify radical expressions.\r\n\r\n<iframe src=\"https:\/\/www.youtube.com\/embed\/CfxhFRHUq_M?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n<h2>Summary<\/h2>\r\nA radical expression is a mathematical way of representing the <i>n<\/i>th root of a number. Square roots and cube roots are the most common radicals, but a root can be any number. To simplify radical expressions, look for exponential factors within the radical, and then use the property [latex] \\sqrt[n]{{{x}^{n}}}=x[\/latex] if <i>n<\/i> is odd and [latex] \\sqrt[n]{{{x}^{n}}}=\\left| x \\right|[\/latex] if <i>n<\/i> is even to pull out quantities. All rules of integer operations and exponents apply when simplifying radical expressions.\r\n\r\nThe steps to consider when simplifying a radical are outlined below.\r\n<div class=\"textbox shaded\">\r\n<h3>Simplifying a radical<\/h3>\r\nWhen working with exponents and radicals:\r\n<ul>\r\n \t<li>If <i>n<\/i> is odd, [latex] \\sqrt[n]{{{x}^{n}}}=x[\/latex].<\/li>\r\n \t<li>If <i>n<\/i> is even, [latex] \\sqrt[n]{{{x}^{n}}}=\\left| x \\right|[\/latex]. (The absolute value accounts for the fact that if <i>x<\/i> is negative and raised to an even power, that number will be positive, as will the <i>n<\/i>th principal root of that number.)<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Multiplying Radicals<\/h2>\r\nYou can do more than just simplify <strong>radical expressions<\/strong>. You can multiply and divide them, too. The product raised to a power rule that we discussed previously will help us find products of radical expressions. Recall the rule:\r\n<div class=\"textbox shaded\">\r\n<h3>A Product Raised to a Power Rule<\/h3>\r\nFor any numbers <i>a<\/i> and <i>b<\/i> and any integer <i>x<\/i>: [latex] {{(ab)}^{x}}={{a}^{x}}\\cdot {{b}^{x}}[\/latex]\r\n\r\nFor any numbers <i>a<\/i> and <i>b<\/i> and any positive integer <i>x<\/i>: [latex] {{(ab)}^{\\frac{1}{x}}}={{a}^{\\frac{1}{x}}}\\cdot {{b}^{\\frac{1}{x}}}[\/latex]\r\n\r\nFor any numbers <i>a<\/i> and <i>b<\/i> and any positive integer <i>x<\/i>: [latex] \\sqrt[x]{ab}=\\sqrt[x]{a}\\cdot \\sqrt[x]{b}[\/latex]\r\n\r\n<\/div>\r\nThe Product Raised to a Power Rule is important because you can use it to multiply radical expressions. Note that you cannot multiply a square root and a cube root using this rule. The indices of the radicals must match in order to multiply them. In our first example, we will work with integers, and then we will move on to expressions with variable radicands.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\sqrt{18}\\cdot \\sqrt{16}[\/latex]\r\n<div class=\"qa-wrapper\" style=\"display: block;\">\r\n\r\n<span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q888021\">Show Solution<\/span>\r\n<div id=\"q888021\" class=\"hidden-answer\" style=\"display: none;\">\r\n\r\nUse the rule [latex] \\sqrt[x]{a}\\cdot \\sqrt[x]{b}=\\sqrt[x]{ab}[\/latex] to multiply the radicands.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\sqrt{18\\cdot 16}\\\\\\sqrt{288}\\end{array}[\/latex]<\/p>\r\nLook for perfect squares in the radicand, and rewrite the radicand as the product of two factors.\r\n<p style=\"text-align: center;\">[latex] \\sqrt{144\\cdot 2}[\/latex]<\/p>\r\nIdentify perfect squares.\r\n<p style=\"text-align: center;\">[latex] \\sqrt{{{(12)}^{2}}\\cdot 2}[\/latex]<\/p>\r\nRewrite as the product of two radicals.\r\n<p style=\"text-align: center;\">[latex] \\sqrt{{{(12)}^{2}}}\\cdot \\sqrt{2}[\/latex]<\/p>\r\nSimplify, using [latex] \\sqrt{{{x}^{2}}}=\\left| x \\right|[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\left| 12 \\right|\\cdot \\sqrt{2}\\\\12\\cdot \\sqrt{2}\\end{array}[\/latex]<\/p>\r\nThe answer is [latex]12\\sqrt{2}[\/latex].\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nYou may have also noticed that both [latex] \\sqrt{18}[\/latex] and [latex] \\sqrt{16}[\/latex] can be written as products involving perfect square factors. How would the expression change if you simplified each radical first, <i>before<\/i> multiplying? In the next example, we will use the same product from above to show that you can simplify before multiplying and get the same result.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\sqrt{18}\\cdot \\sqrt{16}[\/latex]\r\n<div class=\"qa-wrapper\" style=\"display: block;\">\r\n\r\n<span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q479810\">Show Solution<\/span>\r\n<div id=\"q479810\" class=\"hidden-answer\" style=\"display: none;\">\r\n\r\nLook for perfect squares in each radicand, and rewrite as the product of two factors.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{r}\\sqrt{9\\cdot 2}\\cdot \\sqrt{4\\cdot 4}\\\\\\sqrt{3\\cdot 3\\cdot 2}\\cdot \\sqrt{4\\cdot 4}\\end{array}[\/latex]<\/p>\r\nIdentify perfect squares.\r\n<p style=\"text-align: center;\">[latex] \\sqrt{{{(3)}^{2}}\\cdot 2}\\cdot \\sqrt{{{(4)}^{2}}}[\/latex]<\/p>\r\nRewrite as the product of radicals.\r\n<p style=\"text-align: center;\">[latex] \\sqrt{{{(3)}^{2}}}\\cdot \\sqrt{2}\\cdot \\sqrt{{{(4)}^{2}}}[\/latex]<\/p>\r\nSimplify, using [latex] \\sqrt{{{x}^{2}}}=\\left| x \\right|[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\left|3\\right|\\cdot\\sqrt{2}\\cdot\\left|4\\right|\\\\3\\cdot\\sqrt{2}\\cdot4\\end{array}[\/latex]<\/p>\r\nMultiply.\r\n<p style=\"text-align: center;\">[latex]12\\sqrt{2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nIn both cases, you arrive at the same product, [latex] 12\\sqrt{2}[\/latex]. It does not matter whether you multiply the radicands or simplify each radical first.\r\n\r\nYou multiply radical expressions that contain variables in the same manner. As long as the roots of the radical expressions are the same, you can use the Product Raised to a Power Rule to multiply and simplify. Look at the two examples that follow. In both problems, the Product Raised to a Power Rule is used right away and then the expression is simplified. Note that we specify that the variable is non-negative, [latex] x\\ge 0[\/latex], thus allowing us to avoid the need for absolute value.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\sqrt{12{{x}^{4}}}\\cdot \\sqrt{3x^2}[\/latex], [latex] x\\ge 0[\/latex]\r\n<div class=\"qa-wrapper\" style=\"display: block;\">\r\n\r\n<span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q843487\">Show Solution<\/span>\r\n<div id=\"q843487\" class=\"hidden-answer\" style=\"display: none;\">\r\n\r\nUse the rule [latex] \\sqrt[x]{a}\\cdot \\sqrt[x]{b}=\\sqrt[x]{ab}[\/latex] to multiply the radicands.\r\n<p style=\"text-align: center;\">[latex] \\sqrt{12{{x}^{4}}\\cdot 3x^2}\\\\\\sqrt{12\\cdot 3\\cdot {{x}^{4}}\\cdot x^2}[\/latex]<\/p>\r\nRecall that [latex] {{x}^{4}}\\cdot x^2={{x}^{4+2}}[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\sqrt{36\\cdot {{x}^{4+2}}}\\\\\\sqrt{36\\cdot {{x}^{6}}}\\end{array}[\/latex]<\/p>\r\nLook for perfect squares in the radicand.\r\n<p style=\"text-align: center;\">[latex] \\sqrt{{{(6)}^{2}}\\cdot {{({{x}^{3}})}^{2}}}[\/latex]<\/p>\r\nRewrite as the product of radicals.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{c}\\sqrt{{{(6)}^{2}}}\\cdot \\sqrt{{{({{x}^{3}})}^{2}}}\\\\6\\cdot {{x}^{3}}\\end{array}[\/latex]<\/p>\r\nThe answer is [latex]6{{x}^{3}}[\/latex].\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<h3>Analysis of the Solution<\/h3>\r\nEven though our answer contained a variable with an odd exponent that was simplified from an even indexed root, we don\u2019t need to write our answer with absolute value because we specified before we simplified that [latex] x\\ge 0[\/latex]. It is important to read the problem very well when you are doing math. Even the smallest statement like [latex] x\\ge 0[\/latex] can influence the way you write your answer.\r\n\r\nIn our next example, we will multiply two cube roots.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\sqrt[3]{{{x}^{5}}{{y}^{2}}}\\cdot 5\\sqrt[3]{8{{x}^{2}}{{y}^{4}}}[\/latex]\r\n<div class=\"qa-wrapper\" style=\"display: block;\">\r\n\r\n<span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q399955\">Show Solution<\/span>\r\n<div id=\"q399955\" class=\"hidden-answer\" style=\"display: none;\">\r\n\r\nNotice that <i>both<\/i> radicals are cube roots, so you can use the rule [latex] [\/latex] to multiply the radicands.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}5\\sqrt[3]{{{x}^{5}}{{y}^{2}}\\cdot 8{{x}^{2}}{{y}^{4}}}\\\\5\\sqrt[3]{8\\cdot {{x}^{5}}\\cdot {{x}^{2}}\\cdot {{y}^{2}}\\cdot {{y}^{4}}}\\\\5\\sqrt[3]{8\\cdot {{x}^{5+2}}\\cdot {{y}^{2+4}}}\\\\5\\sqrt[3]{8\\cdot {{x}^{7}}\\cdot {{y}^{6}}}\\end{array}[\/latex]<\/p>\r\nLook for perfect cubes in the radicand. Since [latex] {{x}^{7}}[\/latex] is not a perfect cube, it has to be rewritten as [latex] {{x}^{6+1}}={{({{x}^{2}})}^{3}}\\cdot x[\/latex].\r\n<p style=\"text-align: center;\">[latex] 5\\sqrt[3]{{{(2)}^{3}}\\cdot {{({{x}^{2}})}^{3}}\\cdot x\\cdot {{({{y}^{2}})}^{3}}}[\/latex]<\/p>\r\nRewrite as the product of radicals.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{r}5\\sqrt[3]{{{(2)}^{3}}}\\cdot \\sqrt[3]{{{({{x}^{2}})}^{3}}}\\cdot \\sqrt[3]{{{({{y}^{2}})}^{3}}}\\cdot \\sqrt[3]{x}\\\\5\\cdot 2\\cdot {{x}^{2}}\\cdot {{y}^{2}}\\cdot \\sqrt[3]{x}\\end{array}[\/latex]<\/p>\r\nThe answer is [latex]10{{x}^{2}}{{y}^{2}}\\sqrt[3]{x}[\/latex].\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nIn the following video, we present more examples of how to multiply radical expressions.\r\n\r\n<iframe src=\"https:\/\/www.youtube.com\/embed\/PQs10_rFrSM?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n\r\nThis next example is slightly more complicated because there are more than two radicals being multiplied. In this case, notice how the radicals are simplified before multiplication takes place. Remember that the order you choose to use is up to you\u2014you will find that sometimes it is easier to multiply before simplifying, and other times it is easier to simplify before multiplying. With some practice, you may be able to tell which is easier before you approach the problem, but either order will work for all problems.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] 2\\sqrt[4]{16{{x}^{9}}}\\cdot \\sqrt[4]{{{y}^{3}}}\\cdot \\sqrt[4]{81{{x}^{3}}y}[\/latex], [latex] x\\ge 0[\/latex], [latex] y\\ge 0[\/latex]\r\n<div class=\"qa-wrapper\" style=\"display: block;\">\r\n\r\n<span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q257458\">Show Solution<\/span>\r\n<div id=\"q257458\" class=\"hidden-answer\" style=\"display: none;\">\r\n\r\nNotice this expression is multiplying three radicals with the same (fourth) root. Simplify each radical, if possible, before multiplying. Be looking for powers of [latex]4[\/latex] in each radicand.\r\n<p style=\"text-align: center;\">[latex] 2\\sqrt[4]{{{(2)}^{4}}\\cdot {{({{x}^{2}})}^{4}}\\cdot x}\\cdot \\sqrt[4]{{{y}^{3}}}\\cdot \\sqrt[4]{{{(3)}^{4}}\\cdot {{x}^{3}}y}[\/latex]<\/p>\r\nRewrite as the product of radicals.\r\n<p style=\"text-align: center;\">[latex] 2\\sqrt[4]{{{(2)}^{4}}}\\cdot \\sqrt[4]{{{({{x}^{2}})}^{4}}}\\cdot \\sqrt[4]{x}\\cdot \\sqrt[4]{{{y}^{3}}}\\cdot \\sqrt[4]{{{(3)}^{4}}}\\cdot \\sqrt[4]{{{x}^{3}}y}[\/latex]<\/p>\r\nIdentify and pull out powers of [latex]4[\/latex], using the fact that [latex] \\sqrt[4]{{{x}^{4}}}=\\left| x \\right|[\/latex].\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{r}2\\cdot \\left| 2 \\right|\\cdot \\left| {{x}^{2}} \\right|\\cdot \\sqrt[4]{x}\\cdot \\sqrt[4]{{{y}^{3}}}\\cdot \\left| 3 \\right|\\cdot \\sqrt[4]{{{x}^{3}}y}\\\\2\\cdot 2\\cdot {{x}^{2}}\\cdot \\sqrt[4]{x}\\cdot \\sqrt[4]{{{y}^{3}}}\\cdot 3\\cdot \\sqrt[4]{{{x}^{3}}y}\\end{array}[\/latex]<\/p>\r\nSince all the radicals are fourth roots, you can use the rule [latex] \\sqrt[x]{ab}=\\sqrt[x]{a}\\cdot \\sqrt[x]{b}[\/latex] to multiply the radicands.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}2\\cdot 2\\cdot 3\\cdot {{x}^{2}}\\cdot \\sqrt[4]{x\\cdot {{y}^{3}}\\cdot {{x}^{3}}y}\\\\12{{x}^{2}}\\sqrt[4]{{{x}^{1+3}}\\cdot {{y}^{3+1}}}\\end{array}[\/latex]<\/p>\r\nNow that the radicands have been multiplied, look again for powers of [latex]4[\/latex], and pull them out. We can drop the absolute value signs in our final answer because at the start of the problem we were told [latex] x\\ge 0[\/latex], [latex] y\\ge 0[\/latex].\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{l}12{{x}^{2}}\\sqrt[4]{{{x}^{4}}\\cdot {{y}^{4}}}\\\\12{{x}^{2}}\\sqrt[4]{{{x}^{4}}}\\cdot \\sqrt[4]{{{y}^{4}}}\\\\12{{x}^{2}}\\cdot \\left| x \\right|\\cdot \\left| y \\right|\\end{array}[\/latex]<\/p>\r\nThe answer is [latex]12{{x}^{3}}y,\\,\\,x\\ge 0,\\,\\,y\\ge 0[\/latex].\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nIn the following video, we show more examples of multiplying cube roots.\r\n\r\n<iframe src=\"https:\/\/www.youtube.com\/embed\/cxRXofdelIM?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\"><\/iframe>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Simplify expressions with square roots using the order of operations<\/li>\n<li>Simplify expressions with square roots that contain variables<\/li>\n<li>Add and subtract square roots<\/li>\n<li>Simplify Nth roots<\/li>\n<li>Write radicals as rational exponents<\/li>\n<li>Multiply and divide radical expressions<\/li>\n<li>Use the product raised to a power rule to multiply radical expressions<\/li>\n<\/ul>\n<\/div>\n<p>&nbsp;<\/p>\n<h3 data-type=\"title\">Square Roots and the Order of Operations<\/h3>\n<p>When using the order of operations to simplify an expression that has square roots, we treat the radical sign as a grouping symbol. We simplify any expressions under the radical sign before performing other operations.<\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: \u24d0 [latex]\\sqrt{25}+\\sqrt{144}[\/latex] \u24d1 [latex]\\sqrt{25+144}[\/latex].<\/p>\n<p>Solution<\/p>\n<table id=\"eip-id1168466048150\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td>\u24d0 Use the order of operations.<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\sqrt{25}+\\sqrt{144}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify each radical.<\/td>\n<td>[latex]5+12[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add.<\/td>\n<td>[latex]17[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168469785439\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td>\u24d1 Use the order of operations.<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\sqrt{25+144}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add under the radical sign.<\/td>\n<td>[latex]\\sqrt{169}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]13[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146635\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146635&theme=oea&iframe_resize_id=ohm146635&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm146636\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146636&theme=oea&iframe_resize_id=ohm146636&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Notice the different answers in parts \u24d0 and \u24d1 of the example above. It is important to follow the order of operations correctly. In \u24d0, we took each square root first and then added them. In \u24d1, we added under the radical sign first and then found the square root.<\/p>\n<h2 data-type=\"title\">Simplify Variable Expressions with Square Roots<\/h2>\n<p>Expressions with square root that we have looked at so far have not had any variables. What happens when we have to find a square root of a variable expression?<br \/>\nConsider [latex]\\sqrt{9{x}^{2}}[\/latex], where [latex]x\\ge 0[\/latex]. Can you think of an expression whose square is [latex]9{x}^{2}?[\/latex]<\/p>\n<p>[latex]\\begin{array}{ccc}\\hfill {\\left(?\\right)}^{2}& =& 9{x}^{2}\\hfill \\\\ \\hfill {\\left(3x\\right)}^{2}& =& 9{x}^{2}\\text{so}\\sqrt{9{x}^{2}}=3x\\hfill \\end{array}[\/latex]<br \/>\nWhen we use a variable in a square root expression, for our work, we will assume that the variable represents a non-negative number. In every example and exercise that follows, each variable in a square root expression is greater than or equal to zero.<\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]\\sqrt{{x}^{2}}[\/latex], where [latex]x\\ge 0[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q394349\">Show Solution<\/span><\/p>\n<div id=\"q394349\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<br \/>\nThink about what we would have to square to get [latex]{x}^{2}[\/latex] . Algebraically, [latex]{\\left(?\\right)}^{2}={x}^{2}[\/latex]<\/p>\n<table id=\"eip-id1168467419284\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]\\sqrt{{x}^{2}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Since [latex]{\\left(x\\right)}^{2}={x}^{2}[\/latex]<\/td>\n<td>[latex]x[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146637\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146637&theme=oea&iframe_resize_id=ohm146637&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]\\sqrt{16{x}^{2}}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q924940\">Show Solution<\/span><\/p>\n<div id=\"q924940\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168466092245\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]\\sqrt{16{x}^{2}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\text{Since}{\\left(4x\\right)}^{2}=16{x}^{2}[\/latex]<\/td>\n<td>[latex]4x[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146638\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146638&theme=oea&iframe_resize_id=ohm146638&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]-\\sqrt{81{y}^{2}}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q614317\">Show Solution<\/span><\/p>\n<div id=\"q614317\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168468389692\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]-\\sqrt{81{y}^{2}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\text{Since}{\\left(9y\\right)}^{2}=81{y}^{2}[\/latex]<\/td>\n<td>[latex]-9y[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146639\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146639&theme=oea&iframe_resize_id=ohm146639&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]\\sqrt{36{x}^{2}{y}^{2}}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q164861\">Show Solution<\/span><\/p>\n<div id=\"q164861\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168466004603\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]\\sqrt{36{x}^{2}{y}^{2}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\text{Since}{\\left(6xy\\right)}^{2}=36{x}^{2}{y}^{2}[\/latex]<\/td>\n<td>[latex]6xy[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146640\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146640&theme=oea&iframe_resize_id=ohm146640&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<h2>Operations on Square Roots<\/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-1\" 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-2\" 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>Nth Roots and Rational Exponents<\/h2>\n<h3>Using Rational Roots<\/h3>\n<p>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.<br \/>\nSuppose 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>The <em>n<\/em>th root of [latex]a[\/latex] is a number that, when raised to the <em>n<\/em>th power, gives [latex]a[\/latex]. For example, [latex]-3[\/latex] is the 5th root of [latex]-243[\/latex] because [latex]{\\left(-3\\right)}^{5}=-243[\/latex]. If [latex]a[\/latex] is a real number with at least one <em>n<\/em>th root, then the <strong>principal <em>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>n<\/em>th power, equals [latex]a[\/latex].<\/p>\n<p>The principal <em>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\">\n<h3>A General Note: Principal <em>n<\/em>th Root<\/h3>\n<p>If [latex]a[\/latex] is a real number with at least one <em>n<\/em>th root, then the <strong>principal <em>n<\/em>th root<\/strong> 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>n<\/em>th power, equals [latex]a[\/latex]. The <strong>index<\/strong> of the radical is [latex]n[\/latex].<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Simplifying <em>n<\/em>th Roots<\/h3>\n<p>Simplify each of the following:<\/p>\n<ol>\n<li>[latex]\\sqrt[5]{-32}[\/latex]<\/li>\n<li>[latex]\\sqrt[4]{4}\\cdot \\sqrt[4]{1,024}[\/latex]<\/li>\n<li>[latex]-\\sqrt[3]{\\dfrac{8{x}^{6}}{125}}[\/latex]<\/li>\n<li>[latex]8\\sqrt[4]{3}-\\sqrt[4]{48}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q149528\">Show Solution<\/span><\/p>\n<div id=\"q149528\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]\\sqrt[5]{-32}=-2[\/latex] because [latex]{\\left(-2\\right)}^{5}=-32 \\\\ \\text{ }[\/latex]<\/li>\n<li>First, express the product as a single radical expression. [latex]\\sqrt[4]{4\\text{,}096}=8[\/latex] because [latex]{8}^{4}=4,096[\/latex]<\/li>\n<li>[latex]\\begin{align}\\\\ &\\frac{-\\sqrt[3]{8{x}^{6}}}{\\sqrt[3]{125}} && \\text{Write as quotient of two radical expressions}. \\\\ &\\frac{-2{x}^{2}}{5} && \\text{Simplify}. \\\\ \\end{align}[\/latex]<\/li>\n<li>[latex]\\begin{align}\\\\ &8\\sqrt[4]{3}-2\\sqrt[4]{3} && \\text{Simplify to get equal radicands}. \\\\ &6\\sqrt[4]{3} && \\text{Add}. \\end{align}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Simplify.<\/p>\n<ol>\n<li>[latex]\\sqrt[3]{-216}[\/latex]<\/li>\n<li>[latex]\\dfrac{3\\sqrt[4]{80}}{\\sqrt[4]{5}}[\/latex]<\/li>\n<li>[latex]6\\sqrt[3]{9,000}+7\\sqrt[3]{576}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q15987\">Show Solution<\/span><\/p>\n<div id=\"q15987\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]-6[\/latex]<\/li>\n<li>[latex]6[\/latex]<\/li>\n<li>[latex]88\\sqrt[3]{9}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm2564\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2564&#38;theme=oea&#38;iframe_resize_id=ohm2564&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm2565\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2565&#38;theme=oea&#38;iframe_resize_id=ohm2565&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm2567\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2567&#38;theme=oea&#38;iframe_resize_id=ohm2567&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm2592\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2592&#38;theme=oea&#38;iframe_resize_id=ohm2592&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Simplify Radical Expressions<\/h2>\n<p><strong>Radical expressions<\/strong> are expressions that contain radicals. Radical expressions come in many forms, from simple and familiar, such as[latex]\\sqrt{16}[\/latex], to quite complicated, as in [latex]\\sqrt[3]{250{{x}^{4}}y}[\/latex].<\/p>\n<p>To simplify complicated radical expressions, we can use some definitions and rules from simplifying exponents. Recall the <strong>Product Raised to a Power Rule<\/strong> from when you studied exponents. This rule states that the product of two or more non-zero numbers raised to a power is equal to the product of each number raised to the same power. In math terms, it is written [latex]\\left(ab\\right)^{x}=a^{x}\\cdot{b}^{x}[\/latex]. So, for example, you can use the rule to rewrite [latex]{{\\left( 3x \\right)}^{2}}[\/latex] as [latex]{{3}^{2}}\\cdot {{x}^{2}}=9\\cdot {{x}^{2}}=9{{x}^{2}}[\/latex].<\/p>\n<p>Now instead of using the exponent [latex]2[\/latex], use the exponent [latex]\\frac{1}{2}[\/latex]. The exponent is distributed in the same way.<\/p>\n<p style=\"text-align: center;\">[latex]{{\\left( 3x \\right)}^{\\frac{1}{2}}}={{3}^{\\frac{1}{2}}}\\cdot {{x}^{\\frac{1}{2}}}[\/latex]<\/p>\n<p>And since you know that raising a number to the [latex]\\frac{1}{2}[\/latex] power is the same as taking the square root of that number, you can also write it this way.<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt{3x}=\\sqrt{3}\\cdot \\sqrt{x}[\/latex]<\/p>\n<p>Look at that\u2014you can think of any number underneath a radical as the <i>product of separate factors<\/i>, each underneath its own radical.<\/p>\n<div class=\"textbox shaded\">\n<h3>A Product Raised to a Power Rule or sometimes called The Square Root of a Product Rule<\/h3>\n<p>For any real numbers <i>a<\/i> and <i>b<\/i>, [latex]\\sqrt{ab}=\\sqrt{a}\\cdot \\sqrt{b}[\/latex].<\/p>\n<p>For example: [latex]\\sqrt{100}=\\sqrt{10}\\cdot \\sqrt{10}[\/latex], and [latex]\\sqrt{75}=\\sqrt{25}\\cdot \\sqrt{3}[\/latex]<\/p>\n<\/div>\n<p>This rule is important because it helps you think of one radical as the product of multiple radicals. If you can identify perfect squares within a radical, as with [latex]\\sqrt{(2\\cdot 2)(2\\cdot 2)(3\\cdot 3})[\/latex], you can rewrite the expression as the product of multiple perfect squares: [latex]\\sqrt{{{2}^{2}}}\\cdot \\sqrt{{{2}^{2}}}\\cdot \\sqrt{{{3}^{2}}}[\/latex].<\/p>\n<p class=\"p1\">The square root of a product rule will help us simplify roots that are not perfect as is shown the following example.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\sqrt{63}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block;\">\n<p><span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q908978\">Show Solution<\/span><\/p>\n<div id=\"q908978\" class=\"hidden-answer\" style=\"display: none;\">\n<p>[latex]63[\/latex] is not a perfect square so we can use the square root of a product rule to simplify any factors that are perfect squares.<br \/>\nFactor [latex]63[\/latex] into [latex]7[\/latex] and [latex]9[\/latex].<br \/>\n[latex]\\sqrt{7\\cdot 9}[\/latex]<br \/>\n[latex]9[\/latex] is a perfect square, [latex]9=3^2[\/latex], therefore we can rewrite the radicand.<\/p>\n<p>[latex]\\sqrt{7\\cdot {{3}^{2}}}[\/latex]<\/p>\n<p>Using the Product Raised to a Power rule, separate the radical into the product of two factors, each under a radical.<br \/>\n[latex]\\sqrt{7}\\cdot \\sqrt{{{3}^{2}}}[\/latex]<br \/>\nTake the square root of [latex]3^{2}[\/latex].<br \/>\n[latex]\\sqrt{7}\\cdot 3[\/latex]<br \/>\nRearrange factors so the integer appears before the radical and then multiply. This is done so that it is clear that only the [latex]7[\/latex] is under the radical, not the [latex]3[\/latex].<br \/>\n[latex]3\\cdot \\sqrt{7}[\/latex]<br \/>\nThe answer is [latex]3\\sqrt{7}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The final answer [latex]3\\sqrt{7}[\/latex] may look a bit odd, but it is in simplified form. You can read this as \u201cthree radical seven\u201d or \u201cthree times the square root of seven.\u201d<\/p>\n<p>The following video shows more examples of how to simplify square roots that do not have perfect square radicands.<\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/oRd7aBCsmfU?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>Before we move on to simplifying more complex radicals with variables, we need to learn about an important behavior of square roots with variables in the radicand.<\/p>\n<p>Consider the expression [latex]\\sqrt{{{x}^{2}}}[\/latex]. This looks like it should be equal to <i>x<\/i>, right? Test some values for <i>x<\/i> and see what happens.<\/p>\n<p>In the chart below, look along each row and determine whether the value of <i>x<\/i> is the same as the value of [latex]\\sqrt{{{x}^{2}}}[\/latex]. Where are they equal? Where are they not equal?<\/p>\n<p>After doing that for each row, look again and determine whether the value of [latex]\\sqrt{{{x}^{2}}}[\/latex] is the same as the value of [latex]\\left|x\\right|[\/latex].<\/p>\n<table style=\"width: 40%;\">\n<thead>\n<tr style=\"height: 30px;\">\n<th style=\"height: 30px;\">[latex]x[\/latex]<\/th>\n<th style=\"height: 30px;\">[latex]x^{2}[\/latex]<\/th>\n<th style=\"height: 30px;\">[latex]\\sqrt{x^{2}}[\/latex]<\/th>\n<th style=\"height: 30px;\">[latex]\\left|x\\right|[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 30px;\">\n<td style=\"height: 30px;\">[latex]\u22125[\/latex]<\/td>\n<td style=\"height: 30px;\">[latex]25[\/latex]<\/td>\n<td style=\"height: 30px;\">[latex]5[\/latex]<\/td>\n<td style=\"height: 30px;\">[latex]5[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 30px;\">\n<td style=\"height: 30px;\">[latex]\u22122[\/latex]<\/td>\n<td style=\"height: 30px;\">[latex]4[\/latex]<\/td>\n<td style=\"height: 30px;\">[latex]2[\/latex]<\/td>\n<td style=\"height: 30px;\">[latex]2[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\">[latex]0[\/latex]<\/td>\n<td style=\"height: 15px;\">[latex]0[\/latex]<\/td>\n<td style=\"height: 15px;\">[latex]0[\/latex]<\/td>\n<td style=\"height: 15px;\">[latex]0[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\">[latex]6[\/latex]<\/td>\n<td style=\"height: 15px;\">[latex]36[\/latex]<\/td>\n<td style=\"height: 15px;\">[latex]6[\/latex]<\/td>\n<td style=\"height: 15px;\">[latex]6[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15.125px;\">\n<td style=\"height: 15.125px;\">[latex]10[\/latex]<\/td>\n<td style=\"height: 15.125px;\">[latex]100[\/latex]<\/td>\n<td style=\"height: 15.125px;\">[latex]10[\/latex]<\/td>\n<td style=\"height: 15.125px;\">[latex]10[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Notice\u2014in cases where <i>x<\/i> is a negative number, [latex]\\sqrt{x^{2}}\\neq{x}[\/latex]! However, in all cases [latex]\\sqrt{x^{2}}=\\left|x\\right|[\/latex]. You need to consider this fact when simplifying radicals with an even index that contain variables, because by definition [latex]\\sqrt{x^{2}}[\/latex] is always nonnegative.<\/p>\n<div class=\"textbox shaded\">\n<h3>Taking the Square Root of a Radical Expression<\/h3>\n<p>When finding the square root of an expression that contains variables raised to an even power, remember that [latex]\\sqrt{x^{2}}=\\left|x\\right|[\/latex].<\/p>\n<p>Examples: [latex]\\sqrt{9x^{2}}=3\\left|x\\right|[\/latex], and [latex]\\sqrt{16{{x}^{2}}{{y}^{2}}}=4\\left|xy\\right|[\/latex]<\/p>\n<\/div>\n<p>We will combine this with the square root of a product rule in our next example to simplify an expression with three variables in the radicand.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\sqrt{{{a}^{3}}{{b}^{5}}{{c}^{2}}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block;\">\n<p><span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q141094\">Show Solution<\/span><\/p>\n<div id=\"q141094\" class=\"hidden-answer\" style=\"display: none;\">\n<p>Factor to find variables with even exponents.<\/p>\n<p>[latex]\\sqrt{{{a}^{2}}\\cdot a\\cdot {{b}^{4}}\\cdot b\\cdot {{c}^{2}}}[\/latex]<\/p>\n<p>Rewrite [latex]b^{4}[\/latex] as [latex]\\left(b^{2}\\right)^{2}[\/latex].<\/p>\n<p>[latex]\\sqrt{{{a}^{2}}\\cdot a\\cdot {{({{b}^{2}})}^{2}}\\cdot b\\cdot {{c}^{2}}}[\/latex]<\/p>\n<p>Separate the squared factors into individual radicals.<\/p>\n<p>[latex]\\sqrt{{{a}^{2}}}\\cdot \\sqrt{{{({{b}^{2}})}^{2}}}\\cdot \\sqrt{{{c}^{2}}}\\cdot \\sqrt{a\\cdot b}[\/latex]<\/p>\n<p>Take the square root of each radical. Remember that [latex]\\sqrt{{{a}^{2}}}=\\left| a \\right|[\/latex].<\/p>\n<p>[latex]\\left| a \\right|\\cdot {{b}^{2}}\\cdot \\left|{c}\\right|\\cdot \\sqrt{a\\cdot b}[\/latex]<\/p>\n<p>Simplify and multiply.<\/p>\n<p>[latex]\\left| ac \\right|{{b}^{2}}\\sqrt{ab}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h3>Analysis of the Solution<\/h3>\n<p>Why did we not write [latex]b^2[\/latex] as [latex]|b^2|[\/latex]? Because when you square a number, you will always get a positive result, so the principal square root of [latex]\\left(b^2\\right)^2[\/latex] will always be non-negative. One tip for knowing when to apply the absolute value after simplifying any even indexed root is to look at the final exponent on your variable terms. If the exponent is odd \u2013 including [latex]1[\/latex] \u2013 add an absolute value. This applies to simplifying any root with an even index, as we will see in later examples.<\/p>\n<p>In the following video, you will see more examples of how to simplify radical expressions with variables.<\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/q7LqsKPoAKo?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>We will show another example where the simplified expression contains variables with both odd and even powers.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\sqrt{9{{x}^{6}}{{y}^{4}}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block;\">\n<p><span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q41297\">Show Solution<\/span><\/p>\n<div id=\"q41297\" class=\"hidden-answer\" style=\"display: none;\">\n<p>Factor to find identical pairs.<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt{3\\cdot 3\\cdot {{x}^{3}}\\cdot {{x}^{3}}\\cdot {{y}^{2}}\\cdot {{y}^{2}}}[\/latex]<\/p>\n<p>Rewrite the pairs as perfect squares.<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt{{{3}^{2}}\\cdot {{\\left( {{x}^{3}} \\right)}^{2}}\\cdot {{\\left( {{y}^{2}} \\right)}^{2}}}[\/latex]<\/p>\n<p>Separate into individual radicals.<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt{{{3}^{2}}}\\cdot \\sqrt{{{\\left( {{x}^{3}} \\right)}^{2}}}\\cdot \\sqrt{{{\\left( {{y}^{2}} \\right)}^{2}}}[\/latex]<\/p>\n<p>Simplify.<\/p>\n<p style=\"text-align: center;\">[latex]3{{x}^{3}}{{y}^{2}}[\/latex]<\/p>\n<p style=\"text-align: left;\">Because x has an odd power, we will add the absolute value for our final solution.<\/p>\n<p style=\"text-align: center;\">[latex]3|{{x}^{3}}|{{y}^{2}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In our next example, we will start with an expression written with a rational exponent. You will see that you can use a similar process \u2013 factoring and sorting terms into squares \u2013 to simplify this expression.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]{{(36{{x}^{4}})}^{\\frac{1}{2}}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block;\">\n<p><span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q554375\">Show Solution<\/span><\/p>\n<div id=\"q554375\" class=\"hidden-answer\" style=\"display: none;\">\n<p>Rewrite the expression with the fractional exponent as a radical.<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt{36{{x}^{4}}}[\/latex]<\/p>\n<p>Find the square root of both the coefficient and the variable.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r} \\sqrt{{{6}^{2}}\\cdot {{x}^{4}}}\\\\\\sqrt{{{6}^{2}}}\\cdot \\sqrt{{{x}^{4}}}\\\\\\sqrt{{{6}^{2}}}\\cdot \\sqrt{{{({{x}^{2}})}^{2}}}\\\\6\\cdot{x}^{2}\\end{array}[\/latex]<\/p>\n<p>The answer is [latex]6{{x}^{2}}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Here is one more example with perfect squares.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\sqrt{49{{x}^{10}}{{y}^{8}}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block;\">\n<p><span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q283065\">Show Solution<\/span><\/p>\n<div id=\"q283065\" class=\"hidden-answer\" style=\"display: none;\">\n<p>Look for squared numbers and variables. Factor [latex]49[\/latex] into [latex]7\\cdot7[\/latex], [latex]x^{10}[\/latex] into [latex]x^{5}\\cdot{x}^{5}[\/latex], and [latex]y^{8}[\/latex] into [latex]y^{4}\\cdot{y}^{4}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt{7\\cdot 7\\cdot {{x}^{5}}\\cdot {{x}^{5}}\\cdot {{y}^{4}}\\cdot {{y}^{4}}}[\/latex]<\/p>\n<p>Rewrite the pairs as squares.<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt{{{7}^{2}}\\cdot {{({{x}^{5}})}^{2}}\\cdot {{({{y}^{4}})}^{2}}}[\/latex]<\/p>\n<p>Separate the squared factors into individual radicals.<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt{{{7}^{2}}}\\cdot \\sqrt{{{({{x}^{5}})}^{2}}}\\cdot \\sqrt{{{({{y}^{4}})}^{2}}}[\/latex]<\/p>\n<p>Take the square root of each radical using the rule that [latex]\\sqrt{{{x}^{2}}}=x[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]7\\cdot {{x}^{5}}\\cdot {{y}^{4}}[\/latex]<\/p>\n<p>Multiply.<\/p>\n<p style=\"text-align: center;\">[latex]7{{x}^{5}}{{y}^{4}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Simplify Cube Roots<\/h2>\n<p>We can use the same techniques we have used for simplifying square roots to simplify higher order roots. For example, to simplify a cube root, the goal is to find factors under the radical that are perfect cubes so that you can take their cube root. We no longer need to be concerned about whether we have identified the principal root since we are now finding cube roots. Focus on finding identical trios of factors as you simplify.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\sqrt[3]{40{{m}^{5}}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block;\">\n<p><span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q617053\">Show Solution<\/span><\/p>\n<div id=\"q617053\" class=\"hidden-answer\" style=\"display: none;\">\n<p>Factor [latex]40[\/latex] into prime factors.<\/p>\n<p>[latex]\\sqrt[3]{5\\cdot 2\\cdot 2\\cdot 2\\cdot {{m}^{5}}}[\/latex]<\/p>\n<p>Since you are looking for the cube root, you need to find factors that appear [latex]3[\/latex] times under the radical. Rewrite [latex]2\\cdot 2\\cdot 2[\/latex] as [latex]{{2}^{3}}[\/latex].<\/p>\n<p>[latex]\\sqrt[3]{{{2}^{3}}\\cdot 5\\cdot {{m}^{5}}}[\/latex]<\/p>\n<p>Rewrite [latex]{{m}^{5}}[\/latex] as [latex]{{m}^{3}}\\cdot {{m}^{2}}[\/latex].<\/p>\n<p>[latex]\\sqrt[3]{{{2}^{3}}\\cdot 5\\cdot {{m}^{3}}\\cdot {{m}^{2}}}[\/latex]<\/p>\n<p>Rewrite the expression as a product of multiple radicals.<\/p>\n<p>[latex]\\sqrt[3]{{{2}^{3}}}\\cdot \\sqrt[3]{5}\\cdot \\sqrt[3]{{{m}^{3}}}\\cdot \\sqrt[3]{{{m}^{2}}}[\/latex]<\/p>\n<p>Simplify and multiply.<\/p>\n<p>[latex]2\\cdot \\sqrt[3]{5}\\cdot m\\cdot \\sqrt[3]{{{m}^{2}}}[\/latex]<\/p>\n<p>The answer is [latex]2m\\sqrt[3]{5{{m}^{2}}}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Remember that you can take the cube root of a negative expression. In the next example, we will simplify a cube root with a negative radicand.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\sqrt[3]{-27{{x}^{4}}{{y}^{3}}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block;\">\n<p><span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q670300\">Show Solution<\/span><\/p>\n<div id=\"q670300\" class=\"hidden-answer\" style=\"display: none;\">\n<p>Factor the expression into cubes.<\/p>\n<p>Separate the cubed factors into individual radicals.<\/p>\n<p>[latex]\\begin{array}{r}\\sqrt[3]{-1\\cdot 27\\cdot {{x}^{4}}\\cdot {{y}^{3}}}\\\\\\sqrt[3]{{{(-1)}^{3}}\\cdot {{(3)}^{3}}\\cdot {{x}^{3}}\\cdot x\\cdot {{y}^{3}}}\\\\\\sqrt[3]{{{(-1)}^{3}}}\\cdot \\sqrt[3]{{{(3)}^{3}}}\\cdot \\sqrt[3]{{{x}^{3}}}\\cdot \\sqrt[3]{x}\\cdot \\sqrt[3]{{{y}^{3}}}\\end{array}[\/latex]<\/p>\n<p>Simplify the cube roots.<\/p>\n<p>[latex]-1\\cdot 3\\cdot x\\cdot y\\cdot \\sqrt[3]{x}[\/latex]<\/p>\n<p>The answer is [latex]\\sqrt[3]{-27{{x}^{4}}{{y}^{3}}}=-3xy\\sqrt[3]{x}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>You could check your answer by performing the inverse operation. If you are right, when you cube [latex]-3xy\\sqrt[3]{x}[\/latex] you should get [latex]-27{{x}^{4}}{{y}^{3}}[\/latex].<\/p>\n<p>[latex]\\begin{array}{l}\\left( -3xy\\sqrt[3]{x} \\right)\\left( -3xy\\sqrt[3]{x} \\right)\\left( -3xy\\sqrt[3]{x} \\right)\\\\-3\\cdot -3\\cdot -3\\cdot x\\cdot x\\cdot x\\cdot y\\cdot y\\cdot y\\cdot \\sqrt[3]{x}\\cdot \\sqrt[3]{x}\\cdot \\sqrt[3]{x}\\\\-27\\cdot {{x}^{3}}\\cdot {{y}^{3}}\\cdot \\sqrt[3]{{{x}^{3}}}\\\\-27{{x}^{3}}{{y}^{3}}\\cdot x\\\\-27{{x}^{4}}{{y}^{3}}\\end{array}[\/latex]<\/p>\n<p>You can also skip the step of factoring out the negative one once you are comfortable with identifying cubes.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\sqrt[3]{-24{{a}^{5}}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block;\">\n<p><span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q473861\">Show Solution<\/span><\/p>\n<div id=\"q473861\" class=\"hidden-answer\" style=\"display: none;\">\n<p>Factor [latex]\u221224[\/latex] to find perfect cubes. Here, [latex]\u22121[\/latex] and [latex]8[\/latex] are the perfect cubes.<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt[3]{-1\\cdot 8\\cdot 3\\cdot {{a}^{5}}}[\/latex]<\/p>\n<p>Factor variables. You are looking for cube exponents, so you factor [latex]a^{5}[\/latex] into [latex]a^{3}[\/latex] and [latex]a^{2}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt[3]{{{(-1)}^{3}}\\cdot {{2}^{3}}\\cdot 3\\cdot {{a}^{3}}\\cdot {{a}^{2}}}[\/latex]<\/p>\n<p>Separate the factors into individual radicals.<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt[3]{{{(-1)}^{3}}}\\cdot \\sqrt[3]{{{2}^{3}}}\\cdot \\sqrt[3]{{{a}^{3}}}\\cdot \\sqrt[3]{3\\cdot {{a}^{2}}}[\/latex]<\/p>\n<p>Simplify, using the property [latex]\\sqrt[3]{{{x}^{3}}}=x[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]-1\\cdot 2\\cdot a\\cdot \\sqrt[3]{3\\cdot {{a}^{2}}}[\/latex]<\/p>\n<p>This is the simplest form of this expression; all cubes have been pulled out of the radical expression.<\/p>\n<p style=\"text-align: center;\">[latex]-2a\\sqrt[3]{3{{a}^{2}}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video, we show more examples of simplifying cube roots.<\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/BtJruOpmHCE?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Simplifying Fourth Roots<\/h2>\n<p>Now let us move to simplifying fourth degree roots. No matter what root you are simplifying, the same idea applies: find cubes for cube roots, powers of four for fourth roots, etc. Recall that when your simplified expression contains an even indexed radical and a variable factor with an odd exponent, you need to apply an absolute value.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\sqrt[4]{81{{x}^{8}}{{y}^{3}}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block;\">\n<p><span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q295348\">Show Solution<\/span><\/p>\n<div id=\"q295348\" class=\"hidden-answer\" style=\"display: none;\">\n<p>Rewrite the expression.<\/p>\n<p>[latex]\\sqrt[4]{81}\\cdot \\sqrt[4]{{{x}^{8}}}\\cdot \\sqrt[4]{{{y}^{3}}}[\/latex]<\/p>\n<p>Factor each radicand.<\/p>\n<p>[latex]\\sqrt[4]{3\\cdot 3\\cdot 3\\cdot 3}\\cdot \\sqrt[4]{{{x}^{2}}\\cdot {{x}^{2}}\\cdot {{x}^{2}}\\cdot {{x}^{2}}}\\cdot \\sqrt[4]{{{y}^{3}}}[\/latex]<\/p>\n<p>Simplify.<\/p>\n<p>[latex]\\begin{array}{r}\\sqrt[4]{{{3}^{4}}}\\cdot \\sqrt[4]{{{({{x}^{2}})}^{4}}}\\cdot \\sqrt[4]{{{y}^{3}}}\\\\3\\cdot {{x}^{2}}\\cdot \\sqrt[4]{{{y}^{3}}}\\end{array}[\/latex]<\/p>\n<p>The answer is [latex]\\sqrt[4]{81x^{8}y^{3}}=3x^{2}\\sqrt[4]{y^{3}}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>An alternative method to factoring is to rewrite the expression with rational exponents, then use the rules of exponents to simplify. You may find that you prefer one method over the other. Either way, it is nice to have options. We will show the last example again, using this idea.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\sqrt[4]{81{{x}^{8}}{{y}^{3}}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block;\">\n<p><span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q324337\">Show Solution<\/span><\/p>\n<div id=\"q324337\" class=\"hidden-answer\" style=\"display: none;\">\n<p>Rewrite the radical using rational exponents.<\/p>\n<p>[latex]{{(81{{x}^{8}}{{y}^{3}})}^{\\frac{1}{4}}}[\/latex]<\/p>\n<p>Use the rules of exponents to simplify the expression.<\/p>\n<p>[latex]\\begin{array}{r}{{81}^{\\frac{1}{4}}}\\cdot {{x}^{\\frac{8}{4}}}\\cdot {{y}^{\\frac{3}{4}}}\\\\{{(3\\cdot 3\\cdot 3\\cdot 3)}^{\\frac{1}{4}}}{{x}^{2}}{{y}^{\\frac{3}{4}}}\\\\{{({{3}^{4}})}^{\\frac{1}{4}}}{{x}^{2}}{{y}^{\\frac{3}{4}}}\\\\3{{x}^{2}}{{y}^{\\frac{3}{4}}}\\end{array}[\/latex]<\/p>\n<p>Change the expression with the rational exponent back to radical form.<\/p>\n<p>[latex]3{{x}^{2}}\\sqrt[4]{{{y}^{3}}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video, we show another example of how to simplify a fourth and fifth root.<\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/op2LEb0YRyw?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>For our last example, we will simplify a more complicated expression, [latex]\\dfrac{10{{b}^{2}}{{c}^{2}}}{c\\sqrt[3]{8{{b}^{4}}}}[\/latex]<i>.<\/i><i> <\/i>This expression has two variables, a fraction, and a radical. Let us take it step-by-step and see if using fractional exponents can help us simplify it.<br \/>\nWe will start by simplifying the denominator since this is where the radical sign is located. Recall that an exponent in the denominator of a fraction can be rewritten as a negative exponent.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\dfrac{10{{b}^{2}}{{c}^{2}}}{c\\sqrt[3]{8{{b}^{4}}}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block;\">\n<p><span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q962386\">Show Solution<\/span><\/p>\n<div id=\"q962386\" class=\"hidden-answer\" style=\"display: none;\">\n<p>Separate the factors in the denominator.<\/p>\n<p>[latex]\\frac{10{{b}^{2}}{{c}^{2}}}{c\\cdot \\sqrt[3]{8}\\cdot \\sqrt[3]{{{b}^{4}}}}[\/latex]<\/p>\n<p>Take the cube root of [latex]8[\/latex], which is [latex]2[\/latex].<\/p>\n<p>[latex]\\frac{10{{b}^{2}}{{c}^{2}}}{c\\cdot 2\\cdot \\sqrt[3]{{{b}^{4}}}}[\/latex]<\/p>\n<p>Rewrite the radical using a fractional exponent.<\/p>\n<p>[latex]\\frac{10{{b}^{2}}{{c}^{2}}}{c\\cdot 2\\cdot {{b}^{\\frac{4}{3}}}}[\/latex]<\/p>\n<p>Rewrite the fraction as a series of factors in order to cancel factors (see next step).<\/p>\n<p>[latex]\\frac{10}{2}\\cdot \\frac{{{c}^{2}}}{c}\\cdot \\frac{{{b}^{2}}}{{{b}^{\\frac{4}{3}}}}[\/latex]<\/p>\n<p>Simplify the constant and <i>c<\/i> factors.<\/p>\n<p>[latex]5\\cdot c\\cdot \\frac{{{b}^{2}}}{{{b}^{\\frac{4}{3}}}}[\/latex]<\/p>\n<p>Use the rule of negative exponents, <i>n<\/i><sup>\u2013<\/sup><i><sup>x<\/sup><\/i><i>=<\/i>[latex]\\frac{1}{{{n}^{x}}}[\/latex], to rewrite [latex]\\frac{1}{{{b}^{\\tfrac{4}{3}}}}[\/latex] as [latex]{{b}^{-\\tfrac{4}{3}}}[\/latex].<\/p>\n<p>[latex]5c{{b}^{2}}{{b}^{-\\ \\frac{4}{3}}}[\/latex]<\/p>\n<p>Combine the <i>b<\/i> factors by adding the exponents.<\/p>\n<p>[latex]5c{{b}^{\\frac{2}{3}}}[\/latex]<\/p>\n<p>Change the expression with the fractional exponent back to radical form. By convention, an expression is not usually considered simplified if it has a fractional exponent or a radical in the denominator.<\/p>\n<p>[latex]5c\\sqrt[3]{{{b}^{2}}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Well, that took a while, but you did it. You applied what you know about fractional exponents, negative exponents, and the rules of exponents to simplify the expression.<\/p>\n<p>In our last video, we show how to use rational exponents to simplify radical expressions.<\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/CfxhFRHUq_M?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Summary<\/h2>\n<p>A radical expression is a mathematical way of representing the <i>n<\/i>th root of a number. Square roots and cube roots are the most common radicals, but a root can be any number. To simplify radical expressions, look for exponential factors within the radical, and then use the property [latex]\\sqrt[n]{{{x}^{n}}}=x[\/latex] if <i>n<\/i> is odd and [latex]\\sqrt[n]{{{x}^{n}}}=\\left| x \\right|[\/latex] if <i>n<\/i> is even to pull out quantities. All rules of integer operations and exponents apply when simplifying radical expressions.<\/p>\n<p>The steps to consider when simplifying a radical are outlined below.<\/p>\n<div class=\"textbox shaded\">\n<h3>Simplifying a radical<\/h3>\n<p>When working with exponents and radicals:<\/p>\n<ul>\n<li>If <i>n<\/i> is odd, [latex]\\sqrt[n]{{{x}^{n}}}=x[\/latex].<\/li>\n<li>If <i>n<\/i> is even, [latex]\\sqrt[n]{{{x}^{n}}}=\\left| x \\right|[\/latex]. (The absolute value accounts for the fact that if <i>x<\/i> is negative and raised to an even power, that number will be positive, as will the <i>n<\/i>th principal root of that number.)<\/li>\n<\/ul>\n<\/div>\n<h2>Multiplying Radicals<\/h2>\n<p>You can do more than just simplify <strong>radical expressions<\/strong>. You can multiply and divide them, too. The product raised to a power rule that we discussed previously will help us find products of radical expressions. Recall the rule:<\/p>\n<div class=\"textbox shaded\">\n<h3>A Product Raised to a Power Rule<\/h3>\n<p>For any numbers <i>a<\/i> and <i>b<\/i> and any integer <i>x<\/i>: [latex]{{(ab)}^{x}}={{a}^{x}}\\cdot {{b}^{x}}[\/latex]<\/p>\n<p>For any numbers <i>a<\/i> and <i>b<\/i> and any positive integer <i>x<\/i>: [latex]{{(ab)}^{\\frac{1}{x}}}={{a}^{\\frac{1}{x}}}\\cdot {{b}^{\\frac{1}{x}}}[\/latex]<\/p>\n<p>For any numbers <i>a<\/i> and <i>b<\/i> and any positive integer <i>x<\/i>: [latex]\\sqrt[x]{ab}=\\sqrt[x]{a}\\cdot \\sqrt[x]{b}[\/latex]<\/p>\n<\/div>\n<p>The Product Raised to a Power Rule is important because you can use it to multiply radical expressions. Note that you cannot multiply a square root and a cube root using this rule. The indices of the radicals must match in order to multiply them. In our first example, we will work with integers, and then we will move on to expressions with variable radicands.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\sqrt{18}\\cdot \\sqrt{16}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block;\">\n<p><span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q888021\">Show Solution<\/span><\/p>\n<div id=\"q888021\" class=\"hidden-answer\" style=\"display: none;\">\n<p>Use the rule [latex]\\sqrt[x]{a}\\cdot \\sqrt[x]{b}=\\sqrt[x]{ab}[\/latex] to multiply the radicands.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\sqrt{18\\cdot 16}\\\\\\sqrt{288}\\end{array}[\/latex]<\/p>\n<p>Look for perfect squares in the radicand, and rewrite the radicand as the product of two factors.<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt{144\\cdot 2}[\/latex]<\/p>\n<p>Identify perfect squares.<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt{{{(12)}^{2}}\\cdot 2}[\/latex]<\/p>\n<p>Rewrite as the product of two radicals.<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt{{{(12)}^{2}}}\\cdot \\sqrt{2}[\/latex]<\/p>\n<p>Simplify, using [latex]\\sqrt{{{x}^{2}}}=\\left| x \\right|[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\left| 12 \\right|\\cdot \\sqrt{2}\\\\12\\cdot \\sqrt{2}\\end{array}[\/latex]<\/p>\n<p>The answer is [latex]12\\sqrt{2}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>You may have also noticed that both [latex]\\sqrt{18}[\/latex] and [latex]\\sqrt{16}[\/latex] can be written as products involving perfect square factors. How would the expression change if you simplified each radical first, <i>before<\/i> multiplying? In the next example, we will use the same product from above to show that you can simplify before multiplying and get the same result.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\sqrt{18}\\cdot \\sqrt{16}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block;\">\n<p><span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q479810\">Show Solution<\/span><\/p>\n<div id=\"q479810\" class=\"hidden-answer\" style=\"display: none;\">\n<p>Look for perfect squares in each radicand, and rewrite as the product of two factors.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\sqrt{9\\cdot 2}\\cdot \\sqrt{4\\cdot 4}\\\\\\sqrt{3\\cdot 3\\cdot 2}\\cdot \\sqrt{4\\cdot 4}\\end{array}[\/latex]<\/p>\n<p>Identify perfect squares.<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt{{{(3)}^{2}}\\cdot 2}\\cdot \\sqrt{{{(4)}^{2}}}[\/latex]<\/p>\n<p>Rewrite as the product of radicals.<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt{{{(3)}^{2}}}\\cdot \\sqrt{2}\\cdot \\sqrt{{{(4)}^{2}}}[\/latex]<\/p>\n<p>Simplify, using [latex]\\sqrt{{{x}^{2}}}=\\left| x \\right|[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\left|3\\right|\\cdot\\sqrt{2}\\cdot\\left|4\\right|\\\\3\\cdot\\sqrt{2}\\cdot4\\end{array}[\/latex]<\/p>\n<p>Multiply.<\/p>\n<p style=\"text-align: center;\">[latex]12\\sqrt{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In both cases, you arrive at the same product, [latex]12\\sqrt{2}[\/latex]. It does not matter whether you multiply the radicands or simplify each radical first.<\/p>\n<p>You multiply radical expressions that contain variables in the same manner. As long as the roots of the radical expressions are the same, you can use the Product Raised to a Power Rule to multiply and simplify. Look at the two examples that follow. In both problems, the Product Raised to a Power Rule is used right away and then the expression is simplified. Note that we specify that the variable is non-negative, [latex]x\\ge 0[\/latex], thus allowing us to avoid the need for absolute value.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\sqrt{12{{x}^{4}}}\\cdot \\sqrt{3x^2}[\/latex], [latex]x\\ge 0[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block;\">\n<p><span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q843487\">Show Solution<\/span><\/p>\n<div id=\"q843487\" class=\"hidden-answer\" style=\"display: none;\">\n<p>Use the rule [latex]\\sqrt[x]{a}\\cdot \\sqrt[x]{b}=\\sqrt[x]{ab}[\/latex] to multiply the radicands.<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt{12{{x}^{4}}\\cdot 3x^2}\\\\\\sqrt{12\\cdot 3\\cdot {{x}^{4}}\\cdot x^2}[\/latex]<\/p>\n<p>Recall that [latex]{{x}^{4}}\\cdot x^2={{x}^{4+2}}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\sqrt{36\\cdot {{x}^{4+2}}}\\\\\\sqrt{36\\cdot {{x}^{6}}}\\end{array}[\/latex]<\/p>\n<p>Look for perfect squares in the radicand.<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt{{{(6)}^{2}}\\cdot {{({{x}^{3}})}^{2}}}[\/latex]<\/p>\n<p>Rewrite as the product of radicals.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\sqrt{{{(6)}^{2}}}\\cdot \\sqrt{{{({{x}^{3}})}^{2}}}\\\\6\\cdot {{x}^{3}}\\end{array}[\/latex]<\/p>\n<p>The answer is [latex]6{{x}^{3}}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h3>Analysis of the Solution<\/h3>\n<p>Even though our answer contained a variable with an odd exponent that was simplified from an even indexed root, we don\u2019t need to write our answer with absolute value because we specified before we simplified that [latex]x\\ge 0[\/latex]. It is important to read the problem very well when you are doing math. Even the smallest statement like [latex]x\\ge 0[\/latex] can influence the way you write your answer.<\/p>\n<p>In our next example, we will multiply two cube roots.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\sqrt[3]{{{x}^{5}}{{y}^{2}}}\\cdot 5\\sqrt[3]{8{{x}^{2}}{{y}^{4}}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block;\">\n<p><span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q399955\">Show Solution<\/span><\/p>\n<div id=\"q399955\" class=\"hidden-answer\" style=\"display: none;\">\n<p>Notice that <i>both<\/i> radicals are cube roots, so you can use the rule [latex][\/latex] to multiply the radicands.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}5\\sqrt[3]{{{x}^{5}}{{y}^{2}}\\cdot 8{{x}^{2}}{{y}^{4}}}\\\\5\\sqrt[3]{8\\cdot {{x}^{5}}\\cdot {{x}^{2}}\\cdot {{y}^{2}}\\cdot {{y}^{4}}}\\\\5\\sqrt[3]{8\\cdot {{x}^{5+2}}\\cdot {{y}^{2+4}}}\\\\5\\sqrt[3]{8\\cdot {{x}^{7}}\\cdot {{y}^{6}}}\\end{array}[\/latex]<\/p>\n<p>Look for perfect cubes in the radicand. Since [latex]{{x}^{7}}[\/latex] is not a perfect cube, it has to be rewritten as [latex]{{x}^{6+1}}={{({{x}^{2}})}^{3}}\\cdot x[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]5\\sqrt[3]{{{(2)}^{3}}\\cdot {{({{x}^{2}})}^{3}}\\cdot x\\cdot {{({{y}^{2}})}^{3}}}[\/latex]<\/p>\n<p>Rewrite as the product of radicals.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}5\\sqrt[3]{{{(2)}^{3}}}\\cdot \\sqrt[3]{{{({{x}^{2}})}^{3}}}\\cdot \\sqrt[3]{{{({{y}^{2}})}^{3}}}\\cdot \\sqrt[3]{x}\\\\5\\cdot 2\\cdot {{x}^{2}}\\cdot {{y}^{2}}\\cdot \\sqrt[3]{x}\\end{array}[\/latex]<\/p>\n<p>The answer is [latex]10{{x}^{2}}{{y}^{2}}\\sqrt[3]{x}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video, we present more examples of how to multiply radical expressions.<\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/PQs10_rFrSM?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>This next example is slightly more complicated because there are more than two radicals being multiplied. In this case, notice how the radicals are simplified before multiplication takes place. Remember that the order you choose to use is up to you\u2014you will find that sometimes it is easier to multiply before simplifying, and other times it is easier to simplify before multiplying. With some practice, you may be able to tell which is easier before you approach the problem, but either order will work for all problems.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]2\\sqrt[4]{16{{x}^{9}}}\\cdot \\sqrt[4]{{{y}^{3}}}\\cdot \\sqrt[4]{81{{x}^{3}}y}[\/latex], [latex]x\\ge 0[\/latex], [latex]y\\ge 0[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block;\">\n<p><span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q257458\">Show Solution<\/span><\/p>\n<div id=\"q257458\" class=\"hidden-answer\" style=\"display: none;\">\n<p>Notice this expression is multiplying three radicals with the same (fourth) root. Simplify each radical, if possible, before multiplying. Be looking for powers of [latex]4[\/latex] in each radicand.<\/p>\n<p style=\"text-align: center;\">[latex]2\\sqrt[4]{{{(2)}^{4}}\\cdot {{({{x}^{2}})}^{4}}\\cdot x}\\cdot \\sqrt[4]{{{y}^{3}}}\\cdot \\sqrt[4]{{{(3)}^{4}}\\cdot {{x}^{3}}y}[\/latex]<\/p>\n<p>Rewrite as the product of radicals.<\/p>\n<p style=\"text-align: center;\">[latex]2\\sqrt[4]{{{(2)}^{4}}}\\cdot \\sqrt[4]{{{({{x}^{2}})}^{4}}}\\cdot \\sqrt[4]{x}\\cdot \\sqrt[4]{{{y}^{3}}}\\cdot \\sqrt[4]{{{(3)}^{4}}}\\cdot \\sqrt[4]{{{x}^{3}}y}[\/latex]<\/p>\n<p>Identify and pull out powers of [latex]4[\/latex], using the fact that [latex]\\sqrt[4]{{{x}^{4}}}=\\left| x \\right|[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}2\\cdot \\left| 2 \\right|\\cdot \\left| {{x}^{2}} \\right|\\cdot \\sqrt[4]{x}\\cdot \\sqrt[4]{{{y}^{3}}}\\cdot \\left| 3 \\right|\\cdot \\sqrt[4]{{{x}^{3}}y}\\\\2\\cdot 2\\cdot {{x}^{2}}\\cdot \\sqrt[4]{x}\\cdot \\sqrt[4]{{{y}^{3}}}\\cdot 3\\cdot \\sqrt[4]{{{x}^{3}}y}\\end{array}[\/latex]<\/p>\n<p>Since all the radicals are fourth roots, you can use the rule [latex]\\sqrt[x]{ab}=\\sqrt[x]{a}\\cdot \\sqrt[x]{b}[\/latex] to multiply the radicands.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}2\\cdot 2\\cdot 3\\cdot {{x}^{2}}\\cdot \\sqrt[4]{x\\cdot {{y}^{3}}\\cdot {{x}^{3}}y}\\\\12{{x}^{2}}\\sqrt[4]{{{x}^{1+3}}\\cdot {{y}^{3+1}}}\\end{array}[\/latex]<\/p>\n<p>Now that the radicands have been multiplied, look again for powers of [latex]4[\/latex], and pull them out. We can drop the absolute value signs in our final answer because at the start of the problem we were told [latex]x\\ge 0[\/latex], [latex]y\\ge 0[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}12{{x}^{2}}\\sqrt[4]{{{x}^{4}}\\cdot {{y}^{4}}}\\\\12{{x}^{2}}\\sqrt[4]{{{x}^{4}}}\\cdot \\sqrt[4]{{{y}^{4}}}\\\\12{{x}^{2}}\\cdot \\left| x \\right|\\cdot \\left| y \\right|\\end{array}[\/latex]<\/p>\n<p>The answer is [latex]12{{x}^{3}}y,\\,\\,x\\ge 0,\\,\\,y\\ge 0[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video, we show more examples of multiplying cube roots.<\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/cxRXofdelIM?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\"><\/iframe><\/p>\n","protected":false},"author":264444,"menu_order":7,"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-18235","chapter","type-chapter","status-publish","hentry"],"part":18142,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/18235","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":14,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/18235\/revisions"}],"predecessor-version":[{"id":18702,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/18235\/revisions\/18702"}],"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\/18235\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/media?parent=18235"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapter-type?post=18235"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/contributor?post=18235"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/license?post=18235"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}