{"id":1649,"date":"2016-06-22T13:20:42","date_gmt":"2016-06-22T13:20:42","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/?post_type=chapter&#038;p=1649"},"modified":"2019-07-24T21:27:38","modified_gmt":"2019-07-24T21:27:38","slug":"read-or-watch-squares-cubes-and-beyond","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/chapter\/read-or-watch-squares-cubes-and-beyond\/","title":{"raw":"Simplify Radical Expressions","rendered":"Simplify Radical Expressions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Simplify radical expressions using factoring<\/li>\r\n \t<li>Simplify radical expressions\u00a0using rational exponents\u00a0and the laws of exponents<\/li>\r\n \t<li>Define [latex]\\sqrt{x^2}=|x|[\/latex] and apply it when simplifying radical expressions<\/li>\r\n<\/ul>\r\n<\/div>\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].\u00a0So, 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\u00a0[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\u00a0or sometimes called\u00a0The 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[reveal-answer q=\"908978\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"908978\"]\r\n\r\n[latex]63[\/latex] is not a perfect square so we can use the\u00a0square root of a product rule to simplify any factors that are perfect squares.\r\nFactor\u00a0[latex]63[\/latex] into\u00a0[latex]7[\/latex] and\u00a0[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\u00a0[latex]7[\/latex] is under the radical, not the\u00a0[latex]3[\/latex].\r\n[latex] 3\\cdot \\sqrt{7}[\/latex]\r\nThe answer is [latex]3\\sqrt{7}[\/latex].\r\n[\/hidden-answer]\r\n\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\nhttps:\/\/youtu.be\/oRd7aBCsmfU\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].\u00a0You need to consider this fact when simplifying radicals with an even index that contain variables, because by definition [latex]\\sqrt{x^{2}}[\/latex]\u00a0is 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\r\n[reveal-answer q=\"141094\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"141094\"]\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]\u00a0as [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[\/hidden-answer]\r\n\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]? \u00a0Because when you square a number, you will always get a positive result, so the principal square root of\u00a0[latex]\\left(b^2\\right)^2[\/latex] will always be non-negative. One tip for\u00a0knowing when to apply the absolute value after simplifying any even indexed root is to look at the final exponent on your variable terms. \u00a0If the exponent is odd - including\u00a0[latex]1[\/latex] - 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\nhttps:\/\/youtu.be\/q7LqsKPoAKo\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\r\n[reveal-answer q=\"41297\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"41297\"]\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[\/hidden-answer]\r\n\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 - factoring and sorting terms into squares - 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\r\n[reveal-answer q=\"554375\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"554375\"]\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[\/hidden-answer]\r\n\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\r\n[reveal-answer q=\"283065\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"283065\"]\r\n\r\nLook for squared numbers and variables. Factor\u00a0[latex]49[\/latex] into [latex]7\\cdot7[\/latex], [latex]x^{10}[\/latex]\u00a0into [latex]x^{5}\\cdot{x}^{5}[\/latex], and [latex]y^{8}[\/latex]\u00a0into [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[\/hidden-answer]\r\n\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\u00a0so that you can take their cube\u00a0root. 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\u00a0of 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\r\n[reveal-answer q=\"617053\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"617053\"]\r\n\r\nFactor\u00a0[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\u00a0[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[\/hidden-answer]\r\n\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\r\n[reveal-answer q=\"670300\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"670300\"]\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[\/hidden-answer]\r\n\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\r\n[reveal-answer q=\"473861\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"473861\"]\r\n\r\nFactor [latex]\u221224[\/latex] to find perfect cubes. Here, [latex]\u22121[\/latex] and\u00a0[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\u00a0for cube exponents, so you factor\u00a0[latex]a^{5}[\/latex]\u00a0into [latex]a^{3}[\/latex]\u00a0and [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].<em>\u00a0<\/em>\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[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video, we show more examples of simplifying cube roots.\r\n\r\nhttps:\/\/youtu.be\/BtJruOpmHCE\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\r\n[reveal-answer q=\"295348\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"295348\"]\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[\/hidden-answer]\r\n\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\r\n[reveal-answer q=\"324337\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"324337\"]\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[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video, we show another example of how to simplify a fourth and fifth root.\r\n\r\nhttps:\/\/youtu.be\/op2LEb0YRyw\r\n\r\nFor our last example, we will simplify\u00a0a 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\u00a0start 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\r\n[reveal-answer q=\"962386\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"962386\"]\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\u00a0[latex]8[\/latex], which is\u00a0[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,\u00a0<i>n<\/i><sup>-<\/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[\/hidden-answer]\r\n\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\nhttps:\/\/youtu.be\/CfxhFRHUq_M\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>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Simplify radical expressions using factoring<\/li>\n<li>Simplify radical expressions\u00a0using rational exponents\u00a0and the laws of exponents<\/li>\n<li>Define [latex]\\sqrt{x^2}=|x|[\/latex] and apply it when simplifying radical expressions<\/li>\n<\/ul>\n<\/div>\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].\u00a0So, 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\u00a0[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\u00a0or sometimes called\u00a0The 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\"><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\u00a0square root of a product rule to simplify any factors that are perfect squares.<br \/>\nFactor\u00a0[latex]63[\/latex] into\u00a0[latex]7[\/latex] and\u00a0[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\u00a0[latex]7[\/latex] is under the radical, not the\u00a0[latex]3[\/latex].<br \/>\n[latex]3\\cdot \\sqrt{7}[\/latex]<br \/>\nThe answer is [latex]3\\sqrt{7}[\/latex].\n<\/p><\/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\" id=\"oembed-1\" title=\"Simplify Square Roots (Not Perfect Square Radicands)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/oRd7aBCsmfU?feature=oembed&#38;rel=0\" 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].\u00a0You need to consider this fact when simplifying radicals with an even index that contain variables, because by definition [latex]\\sqrt{x^{2}}[\/latex]\u00a0is 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\"><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]\u00a0as [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]? \u00a0Because when you square a number, you will always get a positive result, so the principal square root of\u00a0[latex]\\left(b^2\\right)^2[\/latex] will always be non-negative. One tip for\u00a0knowing when to apply the absolute value after simplifying any even indexed root is to look at the final exponent on your variable terms. \u00a0If the exponent is odd &#8211; including\u00a0[latex]1[\/latex] &#8211; 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\" id=\"oembed-2\" title=\"Simplify Square Roots with Variables\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/q7LqsKPoAKo?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<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\"><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 &#8211; factoring and sorting terms into squares &#8211; 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\"><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\"><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\u00a0[latex]49[\/latex] into [latex]7\\cdot7[\/latex], [latex]x^{10}[\/latex]\u00a0into [latex]x^{5}\\cdot{x}^{5}[\/latex], and [latex]y^{8}[\/latex]\u00a0into [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\u00a0so that you can take their cube\u00a0root. 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\u00a0of 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\"><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\u00a0[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\u00a0[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\"><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\"><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\u00a0[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\u00a0for cube exponents, so you factor\u00a0[latex]a^{5}[\/latex]\u00a0into [latex]a^{3}[\/latex]\u00a0and [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].<em>\u00a0<\/em><\/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\" id=\"oembed-3\" title=\"Simplify Cube Roots (Not Perfect Cube Radicands)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/BtJruOpmHCE?feature=oembed&#38;rel=0\" 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\"><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\"><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\" id=\"oembed-4\" title=\"Simplify Nth Roots with Variables\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/op2LEb0YRyw?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>For our last example, we will simplify\u00a0a 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\u00a0start 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\"><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\u00a0[latex]8[\/latex], which is\u00a0[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,\u00a0<i>n<\/i><sup>&#8211;<\/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\" id=\"oembed-5\" title=\"Simplify Radicals Using Rational Exponents\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/CfxhFRHUq_M?feature=oembed&#38;rel=0\" 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\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1649\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Simplify Square Roots (Not Perfect Square Radicands). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/oRd7aBCsmfU\">https:\/\/youtu.be\/oRd7aBCsmfU<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Simplify Square Roots with Variables. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/q7LqsKPoAKo\">https:\/\/youtu.be\/q7LqsKPoAKo<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Simplify Cube Roots (Not Perfect Cube Radicands). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/BtJruOpmHCE\">https:\/\/youtu.be\/BtJruOpmHCE<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Simplify Nth Roots with Variables. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/op2LEb0YRyw\">https:\/\/youtu.be\/op2LEb0YRyw<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Simplify Radicals Using Rational Exponents. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/CfxhFRHUq_M\">https:\/\/youtu.be\/CfxhFRHUq_M<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: Abramson, Jay. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/li><li>Unit 16: Radical Expressions and Quadratic Equations, from Developmental Math: An Open Program. <strong>Provided by<\/strong>: Monterey Institute of Technology. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/nrocnetwork.org\/dm-opentext\">http:\/\/nrocnetwork.org\/dm-opentext<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":21,"menu_order":5,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Simplify Square Roots (Not Perfect Square Radicands)\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/oRd7aBCsmfU\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Precalculus\",\"author\":\"Abramson, Jay\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at : 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