{"id":4502,"date":"2020-04-13T15:28:34","date_gmt":"2020-04-13T15:28:34","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/mathforlibscoreq\/?post_type=chapter&#038;p=4502"},"modified":"2021-02-06T00:02:35","modified_gmt":"2021-02-06T00:02:35","slug":"identify-and-simplify-roots","status":"web-only","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/chapter\/identify-and-simplify-roots\/","title":{"raw":"Identify and Simplify Roots","rendered":"Identify and Simplify Roots"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Simplify principal square roots using factorization<\/li>\r\n \t<li>Use cube\u00a0root notation to write cube roots<\/li>\r\n \t<li>Simplify\u00a0cube roots using factorization<\/li>\r\n \t<li>Simplify square roots with variables<\/li>\r\n \t<li>Determine when a simplified root needs an absolute value<\/li>\r\n \t<li>Convert\u00a0between radical and exponent notation<\/li>\r\n \t<li>Use the laws of exponents to simplify expressions with rational exponents<\/li>\r\n \t<li>Use rational exponents to simplify radical expressions<\/li>\r\n \t<li style=\"list-style-type: none\"><\/li>\r\n<\/ul>\r\n<\/div>\r\nWe know how to square a number:\r\n\r\n[latex]5^2=25[\/latex] and [latex]\\left(-5\\right)^2=25[\/latex]\r\n\r\nTaking a square root is the opposite of squaring so we can make these statements:\r\n<ul>\r\n \t<li>5 is the nonngeative square root of 25<\/li>\r\n \t<li>-5 is the negative square root of 25<\/li>\r\n<\/ul>\r\nFind the square roots of the following numbers:\r\n<ol>\r\n \t<li>36<\/li>\r\n \t<li>81<\/li>\r\n \t<li>-49<\/li>\r\n \t<li>0<\/li>\r\n<\/ol>\r\n&nbsp;\r\n<ol>\r\n \t<li>We want to find a number whose square is 36. [latex]6^2=36[\/latex] therefore, \u00a0the nonnegative square root of 36 is 6 and the negative square root of 36 is -6<\/li>\r\n \t<li>We want to find a number whose square is 81. [latex]9^2=81[\/latex] therefore, \u00a0the nonnegative square root of 81 is 9 and the negative square root of 81 is -9<\/li>\r\n \t<li>We want to find a number whose square is -49. When you square a real number, the result is always positive. Stop and think about that for a second.\u00a0A negative number times itself is positive, and a positive number times itself is positive. \u00a0Therefore, -49 does not have square roots, there are no real number solutions to this question.<\/li>\r\n \t<li>We want to find a number whose square is 0. [latex]0^2=0[\/latex] therefore, \u00a0the nonnegative square root of 0 is 0. \u00a0We do not assign 0 a sign, so it has only one square root, and that is 0.<\/li>\r\n<\/ol>\r\nThe notation that we use to express a square root for any real number, a, is as follows:\r\n<div class=\"textbox shaded\">\r\n<h4>Writing a Square Root<\/h4>\r\nThe symbol for the square root is called a <strong>radical symbol.<\/strong>\u00a0For a real number, <em>a<\/em> the square root of <em>a<\/em> is written as [latex]\\sqrt{a}[\/latex]\r\n\r\nThe number that is written under the radical symbol is called the <strong>radicand<\/strong>.\r\n\r\nBy definition, the square root symbol, [latex]\\sqrt{\\hphantom{5}}[\/latex] always means to find the nonnegative\u00a0root, called the <strong>principal root<\/strong>.\r\n\r\n[latex]\\sqrt{-a}[\/latex] is not defined, therefore [latex]\\sqrt{a}[\/latex] is defined for [latex]a&gt;0[\/latex]\r\n\r\n<\/div>\r\nLet's do an example similar to\u00a0the example from above, this time using square root notation. \u00a0Note that using the square root notation means that you are only finding the principal root - the nonnegative root.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSimplify\u00a0the following square roots:\r\n<ol>\r\n \t<li>[latex]\\sqrt{16}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{9}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{-9}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{5^2}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"614386\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"614386\"]\r\n<ol>\r\n \t<li>[latex]\\sqrt{16}[\/latex]. \u00a0We are looking for a number whose square is 16, so\u00a0[latex]\\sqrt{16}=4[\/latex]. We only write the nonnegative root because that is how the root symbol is defined.<\/li>\r\n \t<li>[latex]\\sqrt{9}[\/latex]. \u00a0We are looking for a number whose square is 9, so [latex]\\sqrt{9}=3[\/latex].\u00a0We only write the nonnegative root because that is how the root symbol is defined.<\/li>\r\n \t<li>[latex]\\sqrt{-9}[\/latex]. We are looking for a number whose square is -9. \u00a0There are no real numbers whose square is -9, so this radical is not a real number.<\/li>\r\n \t<li>[latex]\\sqrt{5^2}[\/latex]. We are looking for a number whose square is [latex]5^2[\/latex]. \u00a0We already have the number whose square is [latex]5^2[\/latex], it's 5!<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe last problem in the previous example shows us an important relationship between squares and square roots, and we can summarize it as follows:\r\n<div class=\"textbox shaded\">\r\n<h4>\u00a0The square root of a square<\/h4>\r\nFor a nonnegative real number, a, [latex]\\sqrt{a^2}=a[\/latex]\r\n\r\n<\/div>\r\nIn the video that follows, we simplify\u00a0more square roots using the fact that\u00a0\u00a0[latex]\\sqrt{a^2}=a[\/latex] means finding the principal square root.\r\n\r\nhttps:\/\/youtu.be\/B3riJsl7uZM\r\n\r\nWhat if you are working with a number whose square you do not know right away? \u00a0We can use factoring and the product rule for square roots to find square roots such as [latex]\\sqrt{144}[\/latex], or\u00a0\u00a0[latex]\\sqrt{225}[\/latex].\r\n<div class=\"textbox shaded\">\r\n<h4>The Product Rule for Square Roots<\/h4>\r\nGiven that a and b are nonnegative real numbers, [latex]\\sqrt{a\\cdot{b}}=\\sqrt{a}\\cdot\\sqrt{b}[\/latex]\r\n\r\n<\/div>\r\nIn the examples that follow we will bring together these ideas\u00a0to simplify\u00a0square roots of numbers that are not obvious at first glance:\r\n<ul>\r\n \t<li>square root of a square,<\/li>\r\n \t<li>the product rule for square roots<\/li>\r\n \t<li>factoring<\/li>\r\n<\/ul>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify\u00a0[latex] \\sqrt{144}[\/latex]\r\n\r\n[reveal-answer q=\"620082\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"620082\"]\r\n\r\nDetermine the prime factors of 144.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{c}\\sqrt{144}\\\\\\\\\\sqrt{2\\cdot 72}\\\\\\\\\\sqrt{2\\cdot 2\\cdot 36}\\\\\\\\\\sqrt{2\\cdot 2\\cdot 2\\cdot 18}\\\\\\\\\\sqrt{2\\cdot 2\\cdot 2\\cdot 2\\cdot 9}\\\\\\\\\\sqrt{2\\cdot 2\\cdot 2\\cdot 2\\cdot 3\\cdot 3}\\end{array}[\/latex]<\/p>\r\nBecause we are finding a square root, we regroup these factors into squares.\r\n<p style=\"text-align: center\">[latex]\\sqrt{2^2\\cdot 2^2\\cdot3^2}[\/latex]<\/p>\r\nNow we can use the product rule for square roots and the square root of a square idea to finish finding the square root.\r\n\r\n&nbsp;\r\n<p style=\"text-align: center\">[latex]\\begin{array}{c}\\sqrt{2^2\\cdot 2^2\\cdot3^2}\\\\\\\\=\\sqrt{2^2}\\cdot\\sqrt{2^2}\\cdot\\sqrt{3^2}\\\\\\\\=2\\cdot3\\cdot2\\\\\\\\=12\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex] \\sqrt{144}=12[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSimplify [latex]\\sqrt{225}[\/latex]\r\n[reveal-answer q=\"686109\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"686109\"]\r\n\r\nFirst, factor 225:\r\n<p style=\"text-align: center\">[latex]\\begin{array}{c}\\sqrt{225}\\\\\\\\=\\sqrt{5\\cdot45}\\\\\\\\=\\sqrt{5\\cdot5\\cdot9}\\\\\\\\=\\sqrt{5\\cdot5\\cdot3\\cdot3}\\end{array}[\/latex]<\/p>\r\nBecause we are finding a square root, we regroup these factors into squares. Finish simplifying with the product rule for roots, and the square of a square idea.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{c}\\sqrt{5^2\\cdot3^2}\\\\\\\\=\\sqrt{5^2}\\cdot\\sqrt{3^2}\\\\\\\\=5\\cdot3=15\\end{array}[\/latex]<\/p>\r\n\r\n<h4 style=\"text-align: left\">Answer<\/h4>\r\n[latex]\\sqrt{225}=15[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox shaded\">\r\n\r\n<img class=\" wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-300x265.png\" alt=\"Caution\" width=\"49\" height=\"43\" \/>Caution! \u00a0The square root of a product rule applies\u00a0when you have multiplication ONLY under the square root. You cannot apply the rule to sums:\r\n<p style=\"text-align: center\">[latex]\\sqrt{a+b}\\ne\\sqrt{a}+\\sqrt{b}[\/latex]<\/p>\r\n<p style=\"text-align: left\">Prove this to yourself with some real numbers: let a = 64 and b = 36, then use the order of operations to simplify each expression.<\/p>\r\n<p style=\"text-align: center\">[latex]\\begin{array}{c}\\sqrt{64+36}=\\sqrt{100}=10\\\\\\\\\\sqrt{64}+\\sqrt{36}=8+6=14\\\\\\\\10\\ne14\\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\nSo far, you have seen examples that are perfect squares. That is, each is a number whose square root is an integer. But many radical expressions are not perfect squares. Some of these radicals can still be simplified by finding perfect square factors. The example below illustrates how to factor the radicand, looking for pairs of factors that can be expressed as a square.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\sqrt{63}[\/latex]\r\n\r\n[reveal-answer q=\"908978\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"908978\"]Factor 63\r\n<p style=\"text-align: center\">[latex] \\sqrt{7\\cdot 3\\cdot3}[\/latex]<\/p>\r\n<p style=\"text-align: left\">Regroup factors into squares<\/p>\r\n<p style=\"text-align: center\">[latex] \\sqrt{7\\cdot3^2}[\/latex]<\/p>\r\n<p style=\"text-align: left\">Finish simplifying with the product rule for roots, and the square of a square idea.<\/p>\r\n<p style=\"text-align: center\">[latex]\\sqrt{7\\cdot3^2}\\\\\\\\=\\sqrt{7}\\cdot\\sqrt{3^2}\\\\\\\\=\\sqrt{7}\\cdot3[\/latex]<\/p>\r\n<p style=\"text-align: left\">Since 7 is prime and we can't write it as a square, it will have to stay under the radical sign. As a matter of convention, we write the constant, 3, in front of the radical. \u00a0This helps the reader know that the 3 is not under the radical anymore.<\/p>\r\n<p style=\"text-align: center\">[latex] 3\\cdot \\sqrt{7}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex] \\sqrt{63}=3\\sqrt{7}[\/latex]\r\n\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\nIn the next example, we take a bit of a shortcut by making use of the common squares we know, instead of using prime factors. It helps to have the squares of the numbers between 0 and 10 fresh in your mind to make simplifying radicals faster.\r\n<ul>\r\n \t<li style=\"text-align: left\">[latex]0^2=0[\/latex]<\/li>\r\n \t<li style=\"text-align: left\">[latex]2^2=4[\/latex]<\/li>\r\n \t<li style=\"text-align: left\">[latex]3^2=9[\/latex]<\/li>\r\n \t<li style=\"text-align: left\">[latex]4^2=16[\/latex]<\/li>\r\n \t<li style=\"text-align: left\">[latex]5^2=25[\/latex]<\/li>\r\n \t<li style=\"text-align: left\">[latex]6^2=36[\/latex]<\/li>\r\n \t<li style=\"text-align: left\">[latex]7^2=49[\/latex]<\/li>\r\n \t<li style=\"text-align: left\">[latex]8^2=64[\/latex]<\/li>\r\n \t<li style=\"text-align: left\">[latex]9^2=81[\/latex]<\/li>\r\n \t<li style=\"text-align: left\">[latex]10^2=100[\/latex]<\/li>\r\n<\/ul>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\sqrt{2,000}[\/latex]\r\n\r\n[reveal-answer q=\"932245\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"932245\"]Factor 2,000 to find perfect squares.\r\n\r\n[latex] \\begin{array}{r}\\sqrt{100\\cdot 20}\\\\\\\\=\\sqrt{100\\cdot 4\\cdot 5}\\end{array}[\/latex]\r\n\r\n[latex]100=10^2,4=2^2[\/latex]\r\n<p style=\"text-align: center\">[latex]\\begin{array}\\sqrt{100\\cdot 4\\cdot 5}\\\\\\\\= \\sqrt{10^2\\cdot 4^2\\cdot 5}\\\\\\\\=\\sqrt{10^2}\\cdot\\sqrt{4^2}\\cdot\\sqrt{5}\\\\\\\\=10\\cdot4\\cdot\\sqrt{5}\\end{array}[\/latex]<\/p>\r\n&nbsp;\r\n\r\nMultiply.\r\n<p style=\"text-align: center\">[latex] 20\\cdot \\sqrt{5}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex] \\sqrt{2,000}=20\\sqrt{5}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn this last video, we show examples of simplifying radicals that are not perfect squares.\r\n\r\nhttps:\/\/youtu.be\/oRd7aBCsmfU\r\n<h2>Cube Roots<\/h2>\r\nWhile square roots are probably the most common radical, you can also find the third root, the fifth root, the 10th\u00a0root, or really any other <i>n<\/i>th root of a number. Just as the square root is a number that, when squared, gives the radicand, the <strong>cube root<\/strong> is a number that, when cubed, gives the radicand.\r\n\r\nFind the cube roots of the following numbers:\r\n<ol>\r\n \t<li>27<\/li>\r\n \t<li>8<\/li>\r\n \t<li>-8<\/li>\r\n \t<li>0<\/li>\r\n<\/ol>\r\n<ol>\r\n \t<li>We want to find a number whose cube is 27. \u00a0[latex]3\\cdot9=27[\/latex] and [latex]9=3^2[\/latex], so [latex]3\/cdot3\/cdot3=3^3=27[\/latex]<\/li>\r\n \t<li>We want to find a number whose cube is 8. [latex]2\\cdot2\\cdot2=8[\/latex] the cube root of 8 is 2.<\/li>\r\n \t<li>We want to find a number whose cube is -8. We know 2 is the cube root of 8, so maybe we can try -2. [latex]-2\\cdot{-2}\\cdot{-2}=-8[\/latex], so the cube root of -8 is -2. This is different from square roots because multiplying three negative numbers together results in a negative number.<\/li>\r\n \t<li>We want to find a number whose cube is 0. [latex]0\\cdot0\\cdot0[\/latex], no matter how many times you multiply [latex]0[\/latex] by itself, you will always get\u00a0[latex]0[\/latex].<\/li>\r\n<\/ol>\r\nThe cube root of a number is written with a small number 3, called the <strong>index<\/strong>, just outside and above the radical symbol. It looks like [latex] \\sqrt[3]{{}}[\/latex]. This little 3 distinguishes cube roots from square roots which are written without a small number outside and above the radical symbol.\r\n<div class=\"textbox shaded\">\r\n\r\n<img class=\" wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-300x265.png\" alt=\"Caution\" width=\"50\" height=\"44\" \/>Caution! Be careful to distinguish between [latex] \\sqrt[3]{x}[\/latex], the cube root of <i>x<\/i>, and [latex] 3\\sqrt{x}[\/latex], three <i>times<\/i> the <i>square<\/i> root of <i>x<\/i>. They may look similar at first, but they lead you to much different expressions!\r\n\r\n<\/div>\r\nWe can also use factoring to simplify cube roots such as [latex] \\sqrt[3]{125}[\/latex]. You can read this as \u201cthe third root of 125\u201d or \u201cthe cube root of 125.\u201d To simplify this expression, look for a number that, when multiplied by itself two times (for a total of three identical factors), equals 125. Let\u2019s factor 125 and find that number.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\sqrt[3]{125}[\/latex]\r\n\r\n[reveal-answer q=\"517592\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"517592\"]125 ends in 5, so you know that 5 is a factor. Expand 125 into [latex]5\\cdot25[\/latex].\r\n<p style=\"text-align: center\">[latex] \\sqrt[3]{5\\cdot 25}[\/latex]<\/p>\r\nFactor 25 into 5 and 5.\r\n<p style=\"text-align: center\">[latex] \\sqrt[3]{5\\cdot 5\\cdot 5}[\/latex]<\/p>\r\nThe factors are [latex]5\\cdot5\\cdot5[\/latex], or [latex]5^{3}[\/latex].\r\n<p style=\"text-align: center\">[latex] \\sqrt[3]{{{5}^{3}}}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex] \\sqrt[3]{125}=5[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe prime factors of 125 are [latex]5\\cdot5\\cdot5[\/latex], which can be rewritten as [latex]5^{3}[\/latex]. The cube root of a cubed number is the number itself, so [latex] \\sqrt[3]{{{5}^{3}}}=5[\/latex]. You have found the cube root, the three identical factors that when multiplied together give 125. 125 is known as a <strong>perfect cube<\/strong> because its cube root is an integer.\r\n\r\nHere\u2019s an example of how to simplify a radical that is not a perfect cube.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\sqrt[3]{32{{m}^{5}}}[\/latex]\r\n\r\n[reveal-answer q=\"617053\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"617053\"]Factor 32 into prime factors.\r\n<p style=\"text-align: center\">[latex] \\sqrt[3]{2\\cdot 2\\cdot 2\\cdot 2\\cdot 2\\cdot {{m}^{5}}}[\/latex]<\/p>\r\nSince you are looking for the cube root, you need to find factors that appear 3 times under the radical. Rewrite [latex] 2\\cdot 2\\cdot 2[\/latex] as [latex] {{2}^{3}}[\/latex].\r\n<p style=\"text-align: center\">[latex] \\sqrt[3]{{{2}^{3}}\\cdot 2\\cdot 2\\cdot {{m}^{5}}}[\/latex]<\/p>\r\nRewrite [latex] {{m}^{5}}[\/latex] as [latex] {{m}^{3}}\\cdot {{m}^{2}}[\/latex].\r\n<p style=\"text-align: center\">[latex] \\sqrt[3]{{{2}^{3}}\\cdot 2\\cdot 2\\cdot {{m}^{3}}\\cdot {{m}^{2}}}[\/latex]<\/p>\r\nRewrite the expression as a product of multiple radicals.\r\n<p style=\"text-align: center\">[latex] \\sqrt[3]{{{2}^{3}}}\\cdot \\sqrt[3]{2\\cdot 2}\\cdot \\sqrt[3]{{{m}^{3}}}\\cdot \\sqrt[3]{{{m}^{2}}}[\/latex]<\/p>\r\nSimplify and multiply.\r\n<p style=\"text-align: center\">[latex] 2\\cdot \\sqrt[3]{4}\\cdot m\\cdot \\sqrt[3]{{{m}^{2}}}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex] \\sqrt[3]{32{{m}^{5}}}=2m\\sqrt[3]{4{{m}^{2}}}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the example below, we use the following idea:\r\n<p style=\"text-align: center\">\u00a0[latex] \\sqrt[3]{{{(-1)}^{3}}}=-1[\/latex]<\/p>\r\n<p style=\"text-align: left\">\u00a0to simplify the radical. \u00a0You do not have to do this, but it may help you recognize cubes more easily when they are nonnegative.<\/p>\r\n\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\"]Factor the expression into cubes.\r\n\r\nSeparate the cubed factors into individual radicals.\r\n<p style=\"text-align: center\">[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>\r\nSimplify the cube roots.\r\n<p style=\"text-align: center\">[latex] -1\\cdot 3\\cdot x\\cdot y\\cdot \\sqrt[3]{x}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[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\nIn the video that follows, we show more examples if simplifying cube roots.\r\n\r\nhttps:\/\/youtu.be\/9Nh-Ggd2VJo\r\n\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<p style=\"text-align: center\">[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>\r\nYou can find the odd root of a negative number, but you cannot find the even root of a negative number. This means you can simplify the radicals [latex] \\sqrt[3]{-81},\\ \\sqrt[5]{-64}[\/latex], and [latex] \\sqrt[7]{-2187}[\/latex], but you cannot simplify the radicals [latex] \\sqrt[{}]{-100},\\ \\sqrt[4]{-16}[\/latex], or [latex] \\sqrt[6]{-2,500}[\/latex].\r\n\r\nLet\u2019s look at another example.\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\"]Factor [latex]\u221224[\/latex] to find perfect cubes. Here, [latex]\u22121[\/latex] and 8 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\r\n<h4>Answer<\/h4>\r\n[latex] \\sqrt[3]{-24{{a}^{5}}}=-2a\\sqrt[3]{3{{a}^{2}}}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\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<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\sqrt{100{{x}^{2}}{{y}^{4}}}[\/latex]\r\n\r\n[reveal-answer q=\"982628\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"982628\"]Separate factors; look for squared numbers and variables. Factor 100 into [latex]10\\cdot10[\/latex].\r\n<p style=\"text-align: center\">[latex] \\sqrt{10\\cdot 10\\cdot {{x}^{2}}\\cdot {{y}^{4}}}[\/latex]<\/p>\r\nFactor [latex]y^{4}[\/latex]\u00a0into [latex]\\left(y^{2}\\right)^{2}[\/latex].\r\n<p style=\"text-align: center\">[latex] \\sqrt{10\\cdot 10\\cdot {{x}^{2}}\\cdot {{({{y}^{2}})}^{2}}}[\/latex]<\/p>\r\nSeparate the squared factors into individual radicals.\r\n<p style=\"text-align: center\">[latex] \\sqrt{{{10}^{2}}}\\cdot \\sqrt{{{x}^{2}}}\\cdot \\sqrt{{{({{y}^{2}})}^{2}}}[\/latex]<\/p>\r\nTake the square root of each radical . Since you do not know whether <i>x<\/i> is positive or negative, use [latex]\\left|x\\right|[\/latex]\u00a0to account for both possibilities, thereby guaranteeing that your answer will be positive.\r\n<p style=\"text-align: center\">[latex]10\\cdot\\left|x\\right|\\cdot{y}^{2}[\/latex]<\/p>\r\nSimplify and multiply.\r\n<p style=\"text-align: center\">[latex]10\\left|x\\right|y^{2}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex] \\sqrt{100{{x}^{2}}{{y}^{4}}}=10\\left| x \\right|{{y}^{2}}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nYou can check your answer by squaring it to be sure it equals [latex] 100{{x}^{2}}{{y}^{4}}[\/latex].\r\n\r\nIn the last video, we share examples of finding cube roots with negative radicands.\r\n\r\nhttps:\/\/youtu.be\/BtJruOpmHCE\r\n<h2>Simplify Square Roots with Variables<\/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]. Using factoring, you can simplify these radical expressions, too.\r\n<h2 class=\"Subsectiontitleunderline\">Simplifying Square Roots<\/h2>\r\nRadical expressions will sometimes include variables as well as numbers. Consider the expression [latex] \\sqrt{9{{x}^{6}}}[\/latex]. Simplifying a radical expression with\u00a0variables is not as straightforward as the examples we have already shown with integers.\r\n\r\nConsider the expression [latex] \\sqrt{{{x}^{2}}}[\/latex]. This looks like it should be equal to <i>x<\/i>, right? Let\u2019s 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>\r\n<thead>\r\n<tr>\r\n<th>[latex]x[\/latex]<\/th>\r\n<th>[latex]x^{2}[\/latex]<\/th>\r\n<th>[latex]\\sqrt{x^{2}}[\/latex]<\/th>\r\n<th>[latex]\\left|x\\right|[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]\u22125[\/latex]<\/td>\r\n<td>25<\/td>\r\n<td>5<\/td>\r\n<td>5<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\u22122[\/latex]<\/td>\r\n<td>4<\/td>\r\n<td>2<\/td>\r\n<td>2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>0<\/td>\r\n<td>0<\/td>\r\n<td>0<\/td>\r\n<td>0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>6<\/td>\r\n<td>36<\/td>\r\n<td>6<\/td>\r\n<td>6<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>10<\/td>\r\n<td>100<\/td>\r\n<td>10<\/td>\r\n<td>10<\/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]! (This happens because the process of squaring the number loses the negative sign, since a negative times a negative is a positive.) However, in all cases [latex]\\sqrt{x^{2}}=\\left|x\\right|[\/latex].\u00a0You need to consider this fact when simplifying radicals 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 a power, consider 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\nLet\u2019s try it.\r\nThe goal is to find factors under the radical that are perfect squares so that you can take their square root.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\sqrt{9{{x}^{6}}}[\/latex]\r\n\r\n[reveal-answer q=\"41297\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"41297\"]Factor to find identical pairs.\r\n\r\n[latex] \\sqrt{3\\cdot 3\\cdot {{x}^{3}}\\cdot {{x}^{3}}}[\/latex]\r\n\r\nRewrite the pairs as perfect squares, note how we use the power rule for exponents to simplify [latex]x^6[\/latex] into a square: [latex]{x^3}^2[\/latex]\r\n\r\n[latex] \\sqrt{{{3}^{2}}\\cdot {{\\left( {{x}^{3}} \\right)}^{2}}}[\/latex]\r\n\r\nSeparate into individual radicals.\r\n\r\n[latex] \\sqrt{{{3}^{2}}}\\cdot \\sqrt{{{\\left( {{x}^{3}} \\right)}^{2}}}[\/latex]\r\n\r\nSimplify, using the rule that [latex] \\sqrt{{{x}^{2}}}=\\left|x\\right|[\/latex].\r\n\r\n[latex] 3\\left|{{x}^{3}}\\right|[\/latex]\r\n<h4>Answer<\/h4>\r\n[latex] \\sqrt{9{{x}^{6}}}=3\\left|{{x}^{3}}\\right|[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nVariable factors with even exponents can be written as squares. In the example above, [latex] {{x}^{6}}={{x}^{3}}\\cdot{{x}^{3}}={\\left|x^3\\right|}^{2}[\/latex] and\r\n\r\n[latex] {{y}^{4}}={{y}^{2}}\\cdot{{y}^{2}}={\\left(|y^2\\right|)}^{2}[\/latex].\r\n\r\nLet\u2019s try to simplify another radical expression.\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\"]Look for squared numbers and variables. Factor 49 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\r\n[latex] \\sqrt{7\\cdot 7\\cdot {{x}^{5}}\\cdot{{x}^{5}}\\cdot{{y}^{4}}\\cdot{{y}^{4}}}[\/latex]\r\n\r\nRewrite the pairs as squares.\r\n\r\n[latex] \\sqrt{{{7}^{2}}\\cdot{{({{x}^{5}})}^{2}}\\cdot{{({{y}^{4}})}^{2}}}[\/latex]\r\n\r\nSeparate the squared factors into individual radicals.\r\n\r\n[latex] \\sqrt{7^2}\\cdot\\sqrt{({x^5})^2}\\cdot\\sqrt{({y^4})^2}[\/latex]\r\n\r\nTake the square root of each radical using the rule that [latex] \\sqrt{{{x}^{2}}}=\\left|x\\right|[\/latex].\r\n\r\n[latex] 7\\cdot\\left|{{x}^{5}}\\right|\\cdot{{y}^{4}}[\/latex]\r\n\r\nMultiply.\r\n\r\n[latex] 7\\left|{{x}^{5}}\\right|{{y}^{4}}[\/latex]\r\n<h4>Answer<\/h4>\r\n[latex] \\sqrt{49{{x}^{10}}{{y}^{8}}}=7\\left|{{x}^{5}}\\right|{{y}^{4}}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nYou find that the square root of [latex] 49{{x}^{10}}{{y}^{8}}[\/latex] is [latex]7\\left|{{x}^{5}}\\right|{{y}^{4}}[\/latex]. In order to check this calculation, you could square [latex]7\\left|{{x}^{5}}\\right|{{y}^{4}}[\/latex], hoping to arrive at [latex] 49{{x}^{10}}{{y}^{8}}[\/latex]. And, in fact, you would get this expression if you evaluated [latex] {\\left({7\\left|{{x}^{5}}\\right|{{y}^{4}}}\\right)^{2}}[\/latex].\r\n\r\nIn the video that follows we show several examples of simplifying radicals with variables.\r\n\r\nhttps:\/\/youtu.be\/q7LqsKPoAKo\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\"]Factor 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. The entire quantity [latex] a{{b}^{2}}c[\/latex] can be enclosed in the absolute value sign because [latex]b^2[\/latex] will be positive anyway.\r\n\r\n[latex] \\left| a{{b}^{2}}c \\right|\\sqrt{ab}[\/latex]\r\n<h4>Answer<\/h4>\r\n[latex] \\sqrt{{{a}^{3}}{{b}^{5}}{{c}^{2}}}=\\left| a{{b}^{2}}c\\right|\\sqrt{ab}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the next section, we will explore cube roots, and use the methods we have shown here to simplify them. Cube roots are unique from square roots in that it is possible to have a negative number under the root, such as [latex]\\sqrt[3]{-125}[\/latex].\r\n<h2>Rational Exponents<\/h2>\r\nRoots can also be expressed as fractional exponents. \u00a0The square root of a number can be written with\u00a0a\u00a0radical symbol or by raising the number to the [latex] \\frac{1}{2}[\/latex] power. This is illustrated in the table below.\r\n<table style=\"width: 50%\">\r\n<thead>\r\n<tr>\r\n<th>Exponent Form<\/th>\r\n<th>Root Form<\/th>\r\n<th>Root of a Square<\/th>\r\n<th>Simplified<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex] {{25}^{\\frac{1}{2}}}[\/latex]<\/td>\r\n<td>[latex] \\sqrt{25}[\/latex]<\/td>\r\n<td>[latex] \\sqrt{{{5}^{2}}}[\/latex]<\/td>\r\n<td>5<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex] {{16}^{\\frac{1}{2}}}[\/latex]<\/td>\r\n<td>[latex] \\sqrt{16}[\/latex]<\/td>\r\n<td>[latex] \\sqrt{{{4}^{2}}}[\/latex]<\/td>\r\n<td>4<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex] {{100}^{\\frac{1}{2}}}[\/latex]<\/td>\r\n<td>[latex] \\sqrt{100}[\/latex]<\/td>\r\n<td>[latex] \\sqrt{{{10}^{2}}}[\/latex]<\/td>\r\n<td>10<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nUse the example below to familiarize yourself with the different ways to write square roots.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nFill in the missing cells in the table.\r\n<table style=\"width: 50%\">\r\n<thead>\r\n<tr>\r\n<th>Exponent Form<\/th>\r\n<th>Root Form<\/th>\r\n<th>Root of a Square<\/th>\r\n<th>Simplified<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex] {{36}^{\\frac{1}{2}}}[\/latex]<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\sqrt{81}[\/latex]<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td><\/td>\r\n<td>[latex] \\sqrt{{{12}^{2}}}[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"990781\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"990781\"]\r\n<table style=\"width: 50%\">\r\n<thead>\r\n<tr>\r\n<th>Exponent Form<\/th>\r\n<th>Root Form<\/th>\r\n<th>Root of a Square<\/th>\r\n<th>Simplified<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex] {{36}^{\\frac{1}{2}}}[\/latex]<\/td>\r\n<td>[latex] \\sqrt{36}[\/latex]<\/td>\r\n<td>[latex] \\sqrt{{{6}^{2}}}[\/latex]<\/td>\r\n<td>6<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex] {{81}^{\\frac{1}{2}}}[\/latex]<\/td>\r\n<td>[latex] \\sqrt{81}[\/latex]<\/td>\r\n<td>[latex] \\sqrt{{{9}^{2}}}[\/latex]<\/td>\r\n<td>9<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex] {{144}^{\\frac{1}{2}}}[\/latex]<\/td>\r\n<td>[latex] \\sqrt{144}[\/latex]<\/td>\r\n<td>[latex] \\sqrt{{{12}^{2}}}[\/latex]<\/td>\r\n<td>12<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video, we show another example of filling in a table to connect the different notation used for roots.\r\n\r\nhttps:\/\/youtu.be\/eGJgmo2CpN4\r\n\r\nWe can extend the concept of writing [latex]\\sqrt{x}=x^{\\frac{1}{2}}[\/latex] to\u00a0cube roots. Remember, cubing a number raises it to the power of three. Notice that in these examples, the denominator of the rational exponent is the number 3.\r\n<table style=\"width: 50%\">\r\n<thead>\r\n<tr>\r\n<th>\r\n<p style=\"text-align: center\">Radical Form<\/p>\r\n<\/th>\r\n<th>\r\n<p style=\"text-align: center\">Exponent Form<\/p>\r\n<\/th>\r\n<th>\r\n<p style=\"text-align: center\">Integer<\/p>\r\n<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"text-align: center\">[latex] \\sqrt[3]{8}[\/latex]<\/td>\r\n<td style=\"text-align: center\">[latex] {{8}^{\\tfrac{1}{3}}}[\/latex]<\/td>\r\n<td style=\"text-align: center\">2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center\">[latex] \\sqrt[3]{8}[\/latex]<\/td>\r\n<td style=\"text-align: center\">[latex] {{125}^{\\tfrac{1}{3}}}[\/latex]<\/td>\r\n<td style=\"text-align: center\">5<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center\">[latex] \\sqrt[3]{1000}[\/latex]<\/td>\r\n<td style=\"text-align: center\">[latex] {{1000}^{\\tfrac{1}{3}}}[\/latex]<\/td>\r\n<td style=\"text-align: center\">10<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThese examples help us model a relationship between radicals and rational exponents: namely, that the <i>n<\/i>th root of a number can be written as either [latex] \\sqrt[n]{x}[\/latex] or [latex] {{x}^{\\frac{1}{n}}}[\/latex].\r\n<table style=\"width: 50%\">\r\n<thead>\r\n<tr>\r\n<th>\r\n<p style=\"text-align: center\">Radical Form<\/p>\r\n<\/th>\r\n<th>\r\n<p style=\"text-align: center\">Exponent Form<\/p>\r\n<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"text-align: center\">[latex] \\sqrt{x}[\/latex]<\/td>\r\n<td style=\"text-align: center\">[latex] {{x}^{\\tfrac{1}{2}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center\">[latex] \\sqrt[3]{x}[\/latex]<\/td>\r\n<td style=\"text-align: center\">[latex] {{x}^{\\tfrac{1}{3}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center\">[latex] \\sqrt[4]{x}[\/latex]<\/td>\r\n<td style=\"text-align: center\">[latex] {{x}^{\\tfrac{1}{4}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center\">\u2026<\/td>\r\n<td style=\"text-align: center\">\u2026<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center\">[latex] \\sqrt[n]{x}[\/latex]<\/td>\r\n<td style=\"text-align: center\">[latex] {{x}^{\\tfrac{1}{n}}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h2>Convert Between Radical and Exponent Notation<\/h2>\r\nWhen faced with an expression containing a rational exponent, you can rewrite it using a radical. In the table above, notice how the denominator of the rational exponent determines the index of the root. So, an exponent of [latex] \\frac{1}{2}[\/latex] translates to the square root, an exponent of [latex] \\frac{1}{5}[\/latex] translates to the fifth root or [latex] \\sqrt[5]{{\\hphantom{5}}}[\/latex], and [latex] \\frac{1}{8}[\/latex] translates to the eighth root or [latex] \\sqrt[8]{{\\hphantom{5}}}[\/latex] .\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nWrite [latex] \\sqrt[3]{81}[\/latex] as an expression with a rational exponent.\r\n[reveal-answer q=\"612743\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"612743\"]The radical form [latex] \\sqrt[4]{{}}[\/latex] can be rewritten as the exponent [latex] \\frac{1}{4}[\/latex]. Remove the radical and place the exponent next to the base.\r\n<p style=\"text-align: center\">[latex] {{81}^{\\frac{1}{3}}}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex] \\sqrt[3]{81}={{81}^{\\frac{1}{3}}}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nExpress [latex] {{(2x)}^{^{\\frac{1}{3}}}}[\/latex] in radical form.\r\n\r\n[reveal-answer q=\"581351\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"581351\"]Rewrite the expression with the fractional exponent as a radical. The denominator of the fraction determines the root, in this case the cube root.\r\n<p style=\"text-align: center\">[latex]\\sqrt[3]{2x} [\/latex]<\/p>\r\nThe parentheses in [latex] {{\\left( 2x \\right)}^{\\frac{1}{3}}}[\/latex] indicate that the exponent refers to everything within the parentheses.\r\n<h4>Answer<\/h4>\r\n[latex] {{(2x)}^{^{\\frac{1}{3}}}}=\\sqrt[3]{2x}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nRemember that exponents only refer to the quantity immediately to their left unless a grouping symbol is used. The example below looks very similar to the previous example with one important difference\u2014there are no parentheses! Look what happens.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nExpress [latex] 2{{x}^{^{\\frac{1}{2}}}}[\/latex] in radical form.\r\n\r\n[reveal-answer q=\"236347\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"236347\"]Rewrite the expression with the fractional exponent as a radical. The denominator of the fraction determines the root, in this case the cube root.\r\n<p style=\"text-align: center\">[latex] 2\\sqrt{x}[\/latex]<\/p>\r\nThe exponent refers only to the part of the expression immediately to the left of the exponent, in this case <i>x, <\/i>but not the 2.\r\n<h4>Answer<\/h4>\r\n[latex] 2{{x}^{^{\\frac{1}{2}}}}=2\\sqrt{x}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe next example is intended to help you practice\u00a0placing a rational exponent on the appropriate\u00a0terms in an expression that is written in radical form\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nExpress [latex] 4\\sqrt[3]{xy}[\/latex] with rational exponents.\r\n\r\n[reveal-answer q=\"527560\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"527560\"]Rewrite the radical using a rational exponent. The root determines the fraction. In this case, the index of the radical is 3, so the rational exponent will be [latex] \\frac{1}{3}[\/latex].\r\n<p style=\"text-align: center\">[latex] 4{{(xy)}^{\\frac{1}{3}}}[\/latex]<\/p>\r\nSince 4 is outside the radical, it is not included in the grouping symbol and the exponent does not refer to it.\r\n<h4>Answer<\/h4>\r\n[latex] 4\\sqrt[3]{xy}=4{{(xy)}^{\\frac{1}{3}}}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the next video, we show examples of converting between radical and exponent form.\r\n\r\nhttps:\/\/youtu.be\/5cWkVrANBWA\r\n\r\nWhen converting from radical to rational exponent notation, the degree of the root becomes the denominator of the exponent. If you start with a square root, you will have an exponent of [latex]\\frac{1}{2}[\/latex] on the expression in the radical (the radicand). On the other hand, if you start with an exponent of [latex]\\frac{1}{3}[\/latex] you will use a cube root. The following statement summarizes this idea.\r\n<div class=\"textbox shaded\">\r\n<h3>Writing Fractional Exponents<\/h3>\r\nAny radical in the form [latex]\\sqrt[n]{a}[\/latex]\u00a0 can be written using a fractional exponent in the form [latex]a^{\\frac{1}{n}}[\/latex].\r\n\r\n<\/div>\r\n<h2>Simplifying Radical Expressions Using Rational Exponents\u00a0and the Laws of Exponents<\/h2>\r\nLet\u2019s explore some radical expressions now and see how to simplify them. Let's start by simplifying this expression,\u00a0\u00a0[latex] \\sqrt[3]{{{a}^{6}}}[\/latex].\r\n\r\nOne method of simplifying this expression is to factor and pull out groups of [latex]a^{3}[\/latex], as shown below in this example.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\sqrt[3]{{{a}^{6}}}[\/latex]\r\n\r\n[reveal-answer q=\"235013\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"235013\"]Rewrite by factoring out cubes.\r\n<p style=\"text-align: center\">[latex] \\sqrt[3]{{{a}^{3}}\\cdot {{a}^{3}}}[\/latex]<\/p>\r\nWrite each factor under its own radical and simplify.\r\n<p style=\"text-align: center\">[latex] \\begin{array}{r}\\sqrt[3]{{{a}^{3}}}\\cdot \\sqrt[3]{{{a}^{3}}}\\\\a\\cdot{a}\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex] \\sqrt[3]{{{a}^{6}}}={{a}^{2}}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nYou can also simplify this expression by thinking about the radical as an expression with a rational exponent, and using the principle that any radical in the form [latex] \\sqrt[n]{{{a}^{x}}}[\/latex] can be written using a fractional exponent in the form [latex] {{a}^{\\tfrac{x}{n}}}[\/latex].\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\sqrt[3]{{{a}^{6}}}[\/latex]\r\n\r\n[reveal-answer q=\"898415\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"898415\"]Rewrite the radical using a rational exponent.\r\n<p style=\"text-align: center\">[latex] {{a}^{\\frac{6}{3}}}[\/latex]<\/p>\r\nSimplify the exponent.\r\n<p style=\"text-align: center\">[latex] {{a}^{2}}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex] \\sqrt[3]{{{a}^{6}}}={{a}^{2}}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNote that rational exponents are subject to all of the same rules as other exponents when they appear in algebraic expressions.\r\n\r\nBoth simplification methods gave the same result, [latex]a^{2}[\/latex]. Depending on the context of the problem, it may be easier to use one method or the other, but for now, you\u2019ll note that you were able to simplify this expression more quickly using rational exponents than when using the \u201cpull-out\u201d method.\r\n\r\nLet\u2019s try another example.\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\"]Rewrite the radical using rational exponents.\r\n<p style=\"text-align: center\">[latex] {{(81{{x}^{8}}{{y}^{3}})}^{\\frac{1}{4}}}[\/latex]<\/p>\r\nUse the rules of exponents to simplify the expression.\r\n<p style=\"text-align: center\">[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>\r\nChange the expression with the rational exponent back to radical form.\r\n<p style=\"text-align: center\">[latex] 3{{x}^{2}}\\sqrt[4]{{{y}^{3}}}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex] \\sqrt[4]{81{{x}^{8}}{{y}^{3}}}=3{{x}^{2}}\\sqrt[4]{{{y}^{3}}}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nAgain, the alternative method is to work on simplifying under the radical by using factoring. For the example you just solved, it looks like this.\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\"]Rewrite the expression.\r\n<p style=\"text-align: center\">[latex] \\sqrt[4]{81}\\cdot \\sqrt[4]{{{x}^{8}}}\\cdot \\sqrt[4]{{{y}^{3}}}[\/latex]<\/p>\r\nFactor each radicand.\r\n<p style=\"text-align: center\">[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>\r\nSimplify.\r\n<p style=\"text-align: center\">[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>\r\n\r\n<h4>Answer<\/h4>\r\n[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\nThe following video shows more examples of how to simplify a radical expression using rational exponents.\r\n\r\nhttps:\/\/youtu.be\/CfxhFRHUq_M","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Simplify principal square roots using factorization<\/li>\n<li>Use cube\u00a0root notation to write cube roots<\/li>\n<li>Simplify\u00a0cube roots using factorization<\/li>\n<li>Simplify square roots with variables<\/li>\n<li>Determine when a simplified root needs an absolute value<\/li>\n<li>Convert\u00a0between radical and exponent notation<\/li>\n<li>Use the laws of exponents to simplify expressions with rational exponents<\/li>\n<li>Use rational exponents to simplify radical expressions<\/li>\n<li style=\"list-style-type: none\"><\/li>\n<\/ul>\n<\/div>\n<p>We know how to square a number:<\/p>\n<p>[latex]5^2=25[\/latex] and [latex]\\left(-5\\right)^2=25[\/latex]<\/p>\n<p>Taking a square root is the opposite of squaring so we can make these statements:<\/p>\n<ul>\n<li>5 is the nonngeative square root of 25<\/li>\n<li>-5 is the negative square root of 25<\/li>\n<\/ul>\n<p>Find the square roots of the following numbers:<\/p>\n<ol>\n<li>36<\/li>\n<li>81<\/li>\n<li>-49<\/li>\n<li>0<\/li>\n<\/ol>\n<p>&nbsp;<\/p>\n<ol>\n<li>We want to find a number whose square is 36. [latex]6^2=36[\/latex] therefore, \u00a0the nonnegative square root of 36 is 6 and the negative square root of 36 is -6<\/li>\n<li>We want to find a number whose square is 81. [latex]9^2=81[\/latex] therefore, \u00a0the nonnegative square root of 81 is 9 and the negative square root of 81 is -9<\/li>\n<li>We want to find a number whose square is -49. When you square a real number, the result is always positive. Stop and think about that for a second.\u00a0A negative number times itself is positive, and a positive number times itself is positive. \u00a0Therefore, -49 does not have square roots, there are no real number solutions to this question.<\/li>\n<li>We want to find a number whose square is 0. [latex]0^2=0[\/latex] therefore, \u00a0the nonnegative square root of 0 is 0. \u00a0We do not assign 0 a sign, so it has only one square root, and that is 0.<\/li>\n<\/ol>\n<p>The notation that we use to express a square root for any real number, a, is as follows:<\/p>\n<div class=\"textbox shaded\">\n<h4>Writing a Square Root<\/h4>\n<p>The symbol for the square root is called a <strong>radical symbol.<\/strong>\u00a0For a real number, <em>a<\/em> the square root of <em>a<\/em> is written as [latex]\\sqrt{a}[\/latex]<\/p>\n<p>The number that is written under the radical symbol is called the <strong>radicand<\/strong>.<\/p>\n<p>By definition, the square root symbol, [latex]\\sqrt{\\hphantom{5}}[\/latex] always means to find the nonnegative\u00a0root, called the <strong>principal root<\/strong>.<\/p>\n<p>[latex]\\sqrt{-a}[\/latex] is not defined, therefore [latex]\\sqrt{a}[\/latex] is defined for [latex]a>0[\/latex]<\/p>\n<\/div>\n<p>Let&#8217;s do an example similar to\u00a0the example from above, this time using square root notation. \u00a0Note that using the square root notation means that you are only finding the principal root &#8211; the nonnegative root.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Simplify\u00a0the following square roots:<\/p>\n<ol>\n<li>[latex]\\sqrt{16}[\/latex]<\/li>\n<li>[latex]\\sqrt{9}[\/latex]<\/li>\n<li>[latex]\\sqrt{-9}[\/latex]<\/li>\n<li>[latex]\\sqrt{5^2}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q614386\">Show Solution<\/span><\/p>\n<div id=\"q614386\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]\\sqrt{16}[\/latex]. \u00a0We are looking for a number whose square is 16, so\u00a0[latex]\\sqrt{16}=4[\/latex]. We only write the nonnegative root because that is how the root symbol is defined.<\/li>\n<li>[latex]\\sqrt{9}[\/latex]. \u00a0We are looking for a number whose square is 9, so [latex]\\sqrt{9}=3[\/latex].\u00a0We only write the nonnegative root because that is how the root symbol is defined.<\/li>\n<li>[latex]\\sqrt{-9}[\/latex]. We are looking for a number whose square is -9. \u00a0There are no real numbers whose square is -9, so this radical is not a real number.<\/li>\n<li>[latex]\\sqrt{5^2}[\/latex]. We are looking for a number whose square is [latex]5^2[\/latex]. \u00a0We already have the number whose square is [latex]5^2[\/latex], it&#8217;s 5!<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>The last problem in the previous example shows us an important relationship between squares and square roots, and we can summarize it as follows:<\/p>\n<div class=\"textbox shaded\">\n<h4>\u00a0The square root of a square<\/h4>\n<p>For a nonnegative real number, a, [latex]\\sqrt{a^2}=a[\/latex]<\/p>\n<\/div>\n<p>In the video that follows, we simplify\u00a0more square roots using the fact that\u00a0\u00a0[latex]\\sqrt{a^2}=a[\/latex] means finding the principal square root.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Simplify Square Roots (Perfect Square Radicands)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/B3riJsl7uZM?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>What if you are working with a number whose square you do not know right away? \u00a0We can use factoring and the product rule for square roots to find square roots such as [latex]\\sqrt{144}[\/latex], or\u00a0\u00a0[latex]\\sqrt{225}[\/latex].<\/p>\n<div class=\"textbox shaded\">\n<h4>The Product Rule for Square Roots<\/h4>\n<p>Given that a and b are nonnegative real numbers, [latex]\\sqrt{a\\cdot{b}}=\\sqrt{a}\\cdot\\sqrt{b}[\/latex]<\/p>\n<\/div>\n<p>In the examples that follow we will bring together these ideas\u00a0to simplify\u00a0square roots of numbers that are not obvious at first glance:<\/p>\n<ul>\n<li>square root of a square,<\/li>\n<li>the product rule for square roots<\/li>\n<li>factoring<\/li>\n<\/ul>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify\u00a0[latex]\\sqrt{144}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q620082\">Show Solution<\/span><\/p>\n<div id=\"q620082\" class=\"hidden-answer\" style=\"display: none\">\n<p>Determine the prime factors of 144.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{c}\\sqrt{144}\\\\\\\\\\sqrt{2\\cdot 72}\\\\\\\\\\sqrt{2\\cdot 2\\cdot 36}\\\\\\\\\\sqrt{2\\cdot 2\\cdot 2\\cdot 18}\\\\\\\\\\sqrt{2\\cdot 2\\cdot 2\\cdot 2\\cdot 9}\\\\\\\\\\sqrt{2\\cdot 2\\cdot 2\\cdot 2\\cdot 3\\cdot 3}\\end{array}[\/latex]<\/p>\n<p>Because we are finding a square root, we regroup these factors into squares.<\/p>\n<p style=\"text-align: center\">[latex]\\sqrt{2^2\\cdot 2^2\\cdot3^2}[\/latex]<\/p>\n<p>Now we can use the product rule for square roots and the square root of a square idea to finish finding the square root.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{c}\\sqrt{2^2\\cdot 2^2\\cdot3^2}\\\\\\\\=\\sqrt{2^2}\\cdot\\sqrt{2^2}\\cdot\\sqrt{3^2}\\\\\\\\=2\\cdot3\\cdot2\\\\\\\\=12\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\sqrt{144}=12[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Simplify [latex]\\sqrt{225}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q686109\">Show Solution<\/span><\/p>\n<div id=\"q686109\" class=\"hidden-answer\" style=\"display: none\">\n<p>First, factor 225:<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{c}\\sqrt{225}\\\\\\\\=\\sqrt{5\\cdot45}\\\\\\\\=\\sqrt{5\\cdot5\\cdot9}\\\\\\\\=\\sqrt{5\\cdot5\\cdot3\\cdot3}\\end{array}[\/latex]<\/p>\n<p>Because we are finding a square root, we regroup these factors into squares. Finish simplifying with the product rule for roots, and the square of a square idea.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{c}\\sqrt{5^2\\cdot3^2}\\\\\\\\=\\sqrt{5^2}\\cdot\\sqrt{3^2}\\\\\\\\=5\\cdot3=15\\end{array}[\/latex]<\/p>\n<h4 style=\"text-align: left\">Answer<\/h4>\n<p>[latex]\\sqrt{225}=15[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox shaded\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-300x265.png\" alt=\"Caution\" width=\"49\" height=\"43\" \/>Caution! \u00a0The square root of a product rule applies\u00a0when you have multiplication ONLY under the square root. You cannot apply the rule to sums:<\/p>\n<p style=\"text-align: center\">[latex]\\sqrt{a+b}\\ne\\sqrt{a}+\\sqrt{b}[\/latex]<\/p>\n<p style=\"text-align: left\">Prove this to yourself with some real numbers: let a = 64 and b = 36, then use the order of operations to simplify each expression.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{c}\\sqrt{64+36}=\\sqrt{100}=10\\\\\\\\\\sqrt{64}+\\sqrt{36}=8+6=14\\\\\\\\10\\ne14\\end{array}[\/latex]<\/p>\n<\/div>\n<p>So far, you have seen examples that are perfect squares. That is, each is a number whose square root is an integer. But many radical expressions are not perfect squares. Some of these radicals can still be simplified by finding perfect square factors. The example below illustrates how to factor the radicand, looking for pairs of factors that can be expressed as a square.<\/p>\n<div class=\"bcc-box bcc-info\">\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\">Factor 63<\/p>\n<p style=\"text-align: center\">[latex]\\sqrt{7\\cdot 3\\cdot3}[\/latex]<\/p>\n<p style=\"text-align: left\">Regroup factors into squares<\/p>\n<p style=\"text-align: center\">[latex]\\sqrt{7\\cdot3^2}[\/latex]<\/p>\n<p style=\"text-align: left\">Finish simplifying with the product rule for roots, and the square of a square idea.<\/p>\n<p style=\"text-align: center\">[latex]\\sqrt{7\\cdot3^2}\\\\\\\\=\\sqrt{7}\\cdot\\sqrt{3^2}\\\\\\\\=\\sqrt{7}\\cdot3[\/latex]<\/p>\n<p style=\"text-align: left\">Since 7 is prime and we can&#8217;t write it as a square, it will have to stay under the radical sign. As a matter of convention, we write the constant, 3, in front of the radical. \u00a0This helps the reader know that the 3 is not under the radical anymore.<\/p>\n<p style=\"text-align: center\">[latex]3\\cdot \\sqrt{7}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\sqrt{63}=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>In the next example, we take a bit of a shortcut by making use of the common squares we know, instead of using prime factors. It helps to have the squares of the numbers between 0 and 10 fresh in your mind to make simplifying radicals faster.<\/p>\n<ul>\n<li style=\"text-align: left\">[latex]0^2=0[\/latex]<\/li>\n<li style=\"text-align: left\">[latex]2^2=4[\/latex]<\/li>\n<li style=\"text-align: left\">[latex]3^2=9[\/latex]<\/li>\n<li style=\"text-align: left\">[latex]4^2=16[\/latex]<\/li>\n<li style=\"text-align: left\">[latex]5^2=25[\/latex]<\/li>\n<li style=\"text-align: left\">[latex]6^2=36[\/latex]<\/li>\n<li style=\"text-align: left\">[latex]7^2=49[\/latex]<\/li>\n<li style=\"text-align: left\">[latex]8^2=64[\/latex]<\/li>\n<li style=\"text-align: left\">[latex]9^2=81[\/latex]<\/li>\n<li style=\"text-align: left\">[latex]10^2=100[\/latex]<\/li>\n<\/ul>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\sqrt{2,000}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q932245\">Show Solution<\/span><\/p>\n<div id=\"q932245\" class=\"hidden-answer\" style=\"display: none\">Factor 2,000 to find perfect squares.<\/p>\n<p>[latex]\\begin{array}{r}\\sqrt{100\\cdot 20}\\\\\\\\=\\sqrt{100\\cdot 4\\cdot 5}\\end{array}[\/latex]<\/p>\n<p>[latex]100=10^2,4=2^2[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}\\sqrt{100\\cdot 4\\cdot 5}\\\\\\\\= \\sqrt{10^2\\cdot 4^2\\cdot 5}\\\\\\\\=\\sqrt{10^2}\\cdot\\sqrt{4^2}\\cdot\\sqrt{5}\\\\\\\\=10\\cdot4\\cdot\\sqrt{5}\\end{array}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>Multiply.<\/p>\n<p style=\"text-align: center\">[latex]20\\cdot \\sqrt{5}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\sqrt{2,000}=20\\sqrt{5}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In this last video, we show examples of simplifying radicals that are not perfect squares.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" 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<h2>Cube Roots<\/h2>\n<p>While square roots are probably the most common radical, you can also find the third root, the fifth root, the 10th\u00a0root, or really any other <i>n<\/i>th root of a number. Just as the square root is a number that, when squared, gives the radicand, the <strong>cube root<\/strong> is a number that, when cubed, gives the radicand.<\/p>\n<p>Find the cube roots of the following numbers:<\/p>\n<ol>\n<li>27<\/li>\n<li>8<\/li>\n<li>-8<\/li>\n<li>0<\/li>\n<\/ol>\n<ol>\n<li>We want to find a number whose cube is 27. \u00a0[latex]3\\cdot9=27[\/latex] and [latex]9=3^2[\/latex], so [latex]3\/cdot3\/cdot3=3^3=27[\/latex]<\/li>\n<li>We want to find a number whose cube is 8. [latex]2\\cdot2\\cdot2=8[\/latex] the cube root of 8 is 2.<\/li>\n<li>We want to find a number whose cube is -8. We know 2 is the cube root of 8, so maybe we can try -2. [latex]-2\\cdot{-2}\\cdot{-2}=-8[\/latex], so the cube root of -8 is -2. This is different from square roots because multiplying three negative numbers together results in a negative number.<\/li>\n<li>We want to find a number whose cube is 0. [latex]0\\cdot0\\cdot0[\/latex], no matter how many times you multiply [latex]0[\/latex] by itself, you will always get\u00a0[latex]0[\/latex].<\/li>\n<\/ol>\n<p>The cube root of a number is written with a small number 3, called the <strong>index<\/strong>, just outside and above the radical symbol. It looks like [latex]\\sqrt[3]{{}}[\/latex]. This little 3 distinguishes cube roots from square roots which are written without a small number outside and above the radical symbol.<\/p>\n<div class=\"textbox shaded\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-300x265.png\" alt=\"Caution\" width=\"50\" height=\"44\" \/>Caution! Be careful to distinguish between [latex]\\sqrt[3]{x}[\/latex], the cube root of <i>x<\/i>, and [latex]3\\sqrt{x}[\/latex], three <i>times<\/i> the <i>square<\/i> root of <i>x<\/i>. They may look similar at first, but they lead you to much different expressions!<\/p>\n<\/div>\n<p>We can also use factoring to simplify cube roots such as [latex]\\sqrt[3]{125}[\/latex]. You can read this as \u201cthe third root of 125\u201d or \u201cthe cube root of 125.\u201d To simplify this expression, look for a number that, when multiplied by itself two times (for a total of three identical factors), equals 125. Let\u2019s factor 125 and find that number.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\sqrt[3]{125}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q517592\">Show Solution<\/span><\/p>\n<div id=\"q517592\" class=\"hidden-answer\" style=\"display: none\">125 ends in 5, so you know that 5 is a factor. Expand 125 into [latex]5\\cdot25[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\sqrt[3]{5\\cdot 25}[\/latex]<\/p>\n<p>Factor 25 into 5 and 5.<\/p>\n<p style=\"text-align: center\">[latex]\\sqrt[3]{5\\cdot 5\\cdot 5}[\/latex]<\/p>\n<p>The factors are [latex]5\\cdot5\\cdot5[\/latex], or [latex]5^{3}[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\sqrt[3]{{{5}^{3}}}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\sqrt[3]{125}=5[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The prime factors of 125 are [latex]5\\cdot5\\cdot5[\/latex], which can be rewritten as [latex]5^{3}[\/latex]. The cube root of a cubed number is the number itself, so [latex]\\sqrt[3]{{{5}^{3}}}=5[\/latex]. You have found the cube root, the three identical factors that when multiplied together give 125. 125 is known as a <strong>perfect cube<\/strong> because its cube root is an integer.<\/p>\n<p>Here\u2019s an example of how to simplify a radical that is not a perfect cube.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\sqrt[3]{32{{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\">Factor 32 into prime factors.<\/p>\n<p style=\"text-align: center\">[latex]\\sqrt[3]{2\\cdot 2\\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 3 times under the radical. Rewrite [latex]2\\cdot 2\\cdot 2[\/latex] as [latex]{{2}^{3}}[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\sqrt[3]{{{2}^{3}}\\cdot 2\\cdot 2\\cdot {{m}^{5}}}[\/latex]<\/p>\n<p>Rewrite [latex]{{m}^{5}}[\/latex] as [latex]{{m}^{3}}\\cdot {{m}^{2}}[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\sqrt[3]{{{2}^{3}}\\cdot 2\\cdot 2\\cdot {{m}^{3}}\\cdot {{m}^{2}}}[\/latex]<\/p>\n<p>Rewrite the expression as a product of multiple radicals.<\/p>\n<p style=\"text-align: center\">[latex]\\sqrt[3]{{{2}^{3}}}\\cdot \\sqrt[3]{2\\cdot 2}\\cdot \\sqrt[3]{{{m}^{3}}}\\cdot \\sqrt[3]{{{m}^{2}}}[\/latex]<\/p>\n<p>Simplify and multiply.<\/p>\n<p style=\"text-align: center\">[latex]2\\cdot \\sqrt[3]{4}\\cdot m\\cdot \\sqrt[3]{{{m}^{2}}}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\sqrt[3]{32{{m}^{5}}}=2m\\sqrt[3]{4{{m}^{2}}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the example below, we use the following idea:<\/p>\n<p style=\"text-align: center\">\u00a0[latex]\\sqrt[3]{{{(-1)}^{3}}}=-1[\/latex]<\/p>\n<p style=\"text-align: left\">\u00a0to simplify the radical. \u00a0You do not have to do this, but it may help you recognize cubes more easily when they are nonnegative.<\/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\">Factor the expression into cubes.<\/p>\n<p>Separate the cubed factors into individual radicals.<\/p>\n<p style=\"text-align: center\">[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 style=\"text-align: center\">[latex]-1\\cdot 3\\cdot x\\cdot y\\cdot \\sqrt[3]{x}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\sqrt[3]{-27{{x}^{4}}{{y}^{3}}}=-3xy\\sqrt[3]{x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the video that follows, we show more examples if simplifying cube roots.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Simplify Cube Roots (Perfect Cube Radicands)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/9Nh-Ggd2VJo?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\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 style=\"text-align: center\">[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 find the odd root of a negative number, but you cannot find the even root of a negative number. This means you can simplify the radicals [latex]\\sqrt[3]{-81},\\ \\sqrt[5]{-64}[\/latex], and [latex]\\sqrt[7]{-2187}[\/latex], but you cannot simplify the radicals [latex]\\sqrt[{}]{-100},\\ \\sqrt[4]{-16}[\/latex], or [latex]\\sqrt[6]{-2,500}[\/latex].<\/p>\n<p>Let\u2019s look at another example.<\/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\">Factor [latex]\u221224[\/latex] to find perfect cubes. Here, [latex]\u22121[\/latex] and 8 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<h4>Answer<\/h4>\n<p>[latex]\\sqrt[3]{-24{{a}^{5}}}=-2a\\sqrt[3]{3{{a}^{2}}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\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<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\sqrt{100{{x}^{2}}{{y}^{4}}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q982628\">Show Solution<\/span><\/p>\n<div id=\"q982628\" class=\"hidden-answer\" style=\"display: none\">Separate factors; look for squared numbers and variables. Factor 100 into [latex]10\\cdot10[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\sqrt{10\\cdot 10\\cdot {{x}^{2}}\\cdot {{y}^{4}}}[\/latex]<\/p>\n<p>Factor [latex]y^{4}[\/latex]\u00a0into [latex]\\left(y^{2}\\right)^{2}[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\sqrt{10\\cdot 10\\cdot {{x}^{2}}\\cdot {{({{y}^{2}})}^{2}}}[\/latex]<\/p>\n<p>Separate the squared factors into individual radicals.<\/p>\n<p style=\"text-align: center\">[latex]\\sqrt{{{10}^{2}}}\\cdot \\sqrt{{{x}^{2}}}\\cdot \\sqrt{{{({{y}^{2}})}^{2}}}[\/latex]<\/p>\n<p>Take the square root of each radical . Since you do not know whether <i>x<\/i> is positive or negative, use [latex]\\left|x\\right|[\/latex]\u00a0to account for both possibilities, thereby guaranteeing that your answer will be positive.<\/p>\n<p style=\"text-align: center\">[latex]10\\cdot\\left|x\\right|\\cdot{y}^{2}[\/latex]<\/p>\n<p>Simplify and multiply.<\/p>\n<p style=\"text-align: center\">[latex]10\\left|x\\right|y^{2}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\sqrt{100{{x}^{2}}{{y}^{4}}}=10\\left| x \\right|{{y}^{2}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>You can check your answer by squaring it to be sure it equals [latex]100{{x}^{2}}{{y}^{4}}[\/latex].<\/p>\n<p>In the last video, we share examples of finding cube roots with negative radicands.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" 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>Simplify Square Roots with Variables<\/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]. Using factoring, you can simplify these radical expressions, too.<\/p>\n<h2 class=\"Subsectiontitleunderline\">Simplifying Square Roots<\/h2>\n<p>Radical expressions will sometimes include variables as well as numbers. Consider the expression [latex]\\sqrt{9{{x}^{6}}}[\/latex]. Simplifying a radical expression with\u00a0variables is not as straightforward as the examples we have already shown with integers.<\/p>\n<p>Consider the expression [latex]\\sqrt{{{x}^{2}}}[\/latex]. This looks like it should be equal to <i>x<\/i>, right? Let\u2019s 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>\n<thead>\n<tr>\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]x^{2}[\/latex]<\/th>\n<th>[latex]\\sqrt{x^{2}}[\/latex]<\/th>\n<th>[latex]\\left|x\\right|[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]\u22125[\/latex]<\/td>\n<td>25<\/td>\n<td>5<\/td>\n<td>5<\/td>\n<\/tr>\n<tr>\n<td>[latex]\u22122[\/latex]<\/td>\n<td>4<\/td>\n<td>2<\/td>\n<td>2<\/td>\n<\/tr>\n<tr>\n<td>0<\/td>\n<td>0<\/td>\n<td>0<\/td>\n<td>0<\/td>\n<\/tr>\n<tr>\n<td>6<\/td>\n<td>36<\/td>\n<td>6<\/td>\n<td>6<\/td>\n<\/tr>\n<tr>\n<td>10<\/td>\n<td>100<\/td>\n<td>10<\/td>\n<td>10<\/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]! (This happens because the process of squaring the number loses the negative sign, since a negative times a negative is a positive.) However, in all cases [latex]\\sqrt{x^{2}}=\\left|x\\right|[\/latex].\u00a0You need to consider this fact when simplifying radicals 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 a power, consider 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>Let\u2019s try it.<br \/>\nThe goal is to find factors under the radical that are perfect squares so that you can take their square root.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\sqrt{9{{x}^{6}}}[\/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\">Factor to find identical pairs.<\/p>\n<p>[latex]\\sqrt{3\\cdot 3\\cdot {{x}^{3}}\\cdot {{x}^{3}}}[\/latex]<\/p>\n<p>Rewrite the pairs as perfect squares, note how we use the power rule for exponents to simplify [latex]x^6[\/latex] into a square: [latex]{x^3}^2[\/latex]<\/p>\n<p>[latex]\\sqrt{{{3}^{2}}\\cdot {{\\left( {{x}^{3}} \\right)}^{2}}}[\/latex]<\/p>\n<p>Separate into individual radicals.<\/p>\n<p>[latex]\\sqrt{{{3}^{2}}}\\cdot \\sqrt{{{\\left( {{x}^{3}} \\right)}^{2}}}[\/latex]<\/p>\n<p>Simplify, using the rule that [latex]\\sqrt{{{x}^{2}}}=\\left|x\\right|[\/latex].<\/p>\n<p>[latex]3\\left|{{x}^{3}}\\right|[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\sqrt{9{{x}^{6}}}=3\\left|{{x}^{3}}\\right|[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Variable factors with even exponents can be written as squares. In the example above, [latex]{{x}^{6}}={{x}^{3}}\\cdot{{x}^{3}}={\\left|x^3\\right|}^{2}[\/latex] and<\/p>\n<p>[latex]{{y}^{4}}={{y}^{2}}\\cdot{{y}^{2}}={\\left(|y^2\\right|)}^{2}[\/latex].<\/p>\n<p>Let\u2019s try to simplify another radical expression.<\/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\">Look for squared numbers and variables. Factor 49 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>[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>[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>[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}}}=\\left|x\\right|[\/latex].<\/p>\n<p>[latex]7\\cdot\\left|{{x}^{5}}\\right|\\cdot{{y}^{4}}[\/latex]<\/p>\n<p>Multiply.<\/p>\n<p>[latex]7\\left|{{x}^{5}}\\right|{{y}^{4}}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\sqrt{49{{x}^{10}}{{y}^{8}}}=7\\left|{{x}^{5}}\\right|{{y}^{4}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>You find that the square root of [latex]49{{x}^{10}}{{y}^{8}}[\/latex] is [latex]7\\left|{{x}^{5}}\\right|{{y}^{4}}[\/latex]. In order to check this calculation, you could square [latex]7\\left|{{x}^{5}}\\right|{{y}^{4}}[\/latex], hoping to arrive at [latex]49{{x}^{10}}{{y}^{8}}[\/latex]. And, in fact, you would get this expression if you evaluated [latex]{\\left({7\\left|{{x}^{5}}\\right|{{y}^{4}}}\\right)^{2}}[\/latex].<\/p>\n<p>In the video that follows we show several examples of simplifying radicals with variables.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-5\" 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<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\">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. The entire quantity [latex]a{{b}^{2}}c[\/latex] can be enclosed in the absolute value sign because [latex]b^2[\/latex] will be positive anyway.<\/p>\n<p>[latex]\\left| a{{b}^{2}}c \\right|\\sqrt{ab}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\sqrt{{{a}^{3}}{{b}^{5}}{{c}^{2}}}=\\left| a{{b}^{2}}c\\right|\\sqrt{ab}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the next section, we will explore cube roots, and use the methods we have shown here to simplify them. Cube roots are unique from square roots in that it is possible to have a negative number under the root, such as [latex]\\sqrt[3]{-125}[\/latex].<\/p>\n<h2>Rational Exponents<\/h2>\n<p>Roots can also be expressed as fractional exponents. \u00a0The square root of a number can be written with\u00a0a\u00a0radical symbol or by raising the number to the [latex]\\frac{1}{2}[\/latex] power. This is illustrated in the table below.<\/p>\n<table style=\"width: 50%\">\n<thead>\n<tr>\n<th>Exponent Form<\/th>\n<th>Root Form<\/th>\n<th>Root of a Square<\/th>\n<th>Simplified<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]{{25}^{\\frac{1}{2}}}[\/latex]<\/td>\n<td>[latex]\\sqrt{25}[\/latex]<\/td>\n<td>[latex]\\sqrt{{{5}^{2}}}[\/latex]<\/td>\n<td>5<\/td>\n<\/tr>\n<tr>\n<td>[latex]{{16}^{\\frac{1}{2}}}[\/latex]<\/td>\n<td>[latex]\\sqrt{16}[\/latex]<\/td>\n<td>[latex]\\sqrt{{{4}^{2}}}[\/latex]<\/td>\n<td>4<\/td>\n<\/tr>\n<tr>\n<td>[latex]{{100}^{\\frac{1}{2}}}[\/latex]<\/td>\n<td>[latex]\\sqrt{100}[\/latex]<\/td>\n<td>[latex]\\sqrt{{{10}^{2}}}[\/latex]<\/td>\n<td>10<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Use the example below to familiarize yourself with the different ways to write square roots.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Fill in the missing cells in the table.<\/p>\n<table style=\"width: 50%\">\n<thead>\n<tr>\n<th>Exponent Form<\/th>\n<th>Root Form<\/th>\n<th>Root of a Square<\/th>\n<th>Simplified<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]{{36}^{\\frac{1}{2}}}[\/latex]<\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]\\sqrt{81}[\/latex]<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><\/td>\n<td>[latex]\\sqrt{{{12}^{2}}}[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q990781\">Show Solution<\/span><\/p>\n<div id=\"q990781\" class=\"hidden-answer\" style=\"display: none\">\n<table style=\"width: 50%\">\n<thead>\n<tr>\n<th>Exponent Form<\/th>\n<th>Root Form<\/th>\n<th>Root of a Square<\/th>\n<th>Simplified<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]{{36}^{\\frac{1}{2}}}[\/latex]<\/td>\n<td>[latex]\\sqrt{36}[\/latex]<\/td>\n<td>[latex]\\sqrt{{{6}^{2}}}[\/latex]<\/td>\n<td>6<\/td>\n<\/tr>\n<tr>\n<td>[latex]{{81}^{\\frac{1}{2}}}[\/latex]<\/td>\n<td>[latex]\\sqrt{81}[\/latex]<\/td>\n<td>[latex]\\sqrt{{{9}^{2}}}[\/latex]<\/td>\n<td>9<\/td>\n<\/tr>\n<tr>\n<td>[latex]{{144}^{\\frac{1}{2}}}[\/latex]<\/td>\n<td>[latex]\\sqrt{144}[\/latex]<\/td>\n<td>[latex]\\sqrt{{{12}^{2}}}[\/latex]<\/td>\n<td>12<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video, we show another example of filling in a table to connect the different notation used for roots.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-6\" title=\"Write Basic Expression in Radical Form and Using Rational Exponents\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/eGJgmo2CpN4?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>We can extend the concept of writing [latex]\\sqrt{x}=x^{\\frac{1}{2}}[\/latex] to\u00a0cube roots. Remember, cubing a number raises it to the power of three. Notice that in these examples, the denominator of the rational exponent is the number 3.<\/p>\n<table style=\"width: 50%\">\n<thead>\n<tr>\n<th>\n<p style=\"text-align: center\">Radical Form<\/p>\n<\/th>\n<th>\n<p style=\"text-align: center\">Exponent Form<\/p>\n<\/th>\n<th>\n<p style=\"text-align: center\">Integer<\/p>\n<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"text-align: center\">[latex]\\sqrt[3]{8}[\/latex]<\/td>\n<td style=\"text-align: center\">[latex]{{8}^{\\tfrac{1}{3}}}[\/latex]<\/td>\n<td style=\"text-align: center\">2<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center\">[latex]\\sqrt[3]{8}[\/latex]<\/td>\n<td style=\"text-align: center\">[latex]{{125}^{\\tfrac{1}{3}}}[\/latex]<\/td>\n<td style=\"text-align: center\">5<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center\">[latex]\\sqrt[3]{1000}[\/latex]<\/td>\n<td style=\"text-align: center\">[latex]{{1000}^{\\tfrac{1}{3}}}[\/latex]<\/td>\n<td style=\"text-align: center\">10<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>These examples help us model a relationship between radicals and rational exponents: namely, that the <i>n<\/i>th root of a number can be written as either [latex]\\sqrt[n]{x}[\/latex] or [latex]{{x}^{\\frac{1}{n}}}[\/latex].<\/p>\n<table style=\"width: 50%\">\n<thead>\n<tr>\n<th>\n<p style=\"text-align: center\">Radical Form<\/p>\n<\/th>\n<th>\n<p style=\"text-align: center\">Exponent Form<\/p>\n<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"text-align: center\">[latex]\\sqrt{x}[\/latex]<\/td>\n<td style=\"text-align: center\">[latex]{{x}^{\\tfrac{1}{2}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center\">[latex]\\sqrt[3]{x}[\/latex]<\/td>\n<td style=\"text-align: center\">[latex]{{x}^{\\tfrac{1}{3}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center\">[latex]\\sqrt[4]{x}[\/latex]<\/td>\n<td style=\"text-align: center\">[latex]{{x}^{\\tfrac{1}{4}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center\">\u2026<\/td>\n<td style=\"text-align: center\">\u2026<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center\">[latex]\\sqrt[n]{x}[\/latex]<\/td>\n<td style=\"text-align: center\">[latex]{{x}^{\\tfrac{1}{n}}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Convert Between Radical and Exponent Notation<\/h2>\n<p>When faced with an expression containing a rational exponent, you can rewrite it using a radical. In the table above, notice how the denominator of the rational exponent determines the index of the root. So, an exponent of [latex]\\frac{1}{2}[\/latex] translates to the square root, an exponent of [latex]\\frac{1}{5}[\/latex] translates to the fifth root or [latex]\\sqrt[5]{{\\hphantom{5}}}[\/latex], and [latex]\\frac{1}{8}[\/latex] translates to the eighth root or [latex]\\sqrt[8]{{\\hphantom{5}}}[\/latex] .<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Write [latex]\\sqrt[3]{81}[\/latex] as an expression with a rational exponent.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q612743\">Show Solution<\/span><\/p>\n<div id=\"q612743\" class=\"hidden-answer\" style=\"display: none\">The radical form [latex]\\sqrt[4]{{}}[\/latex] can be rewritten as the exponent [latex]\\frac{1}{4}[\/latex]. Remove the radical and place the exponent next to the base.<\/p>\n<p style=\"text-align: center\">[latex]{{81}^{\\frac{1}{3}}}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\sqrt[3]{81}={{81}^{\\frac{1}{3}}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Express [latex]{{(2x)}^{^{\\frac{1}{3}}}}[\/latex] in radical form.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q581351\">Show Solution<\/span><\/p>\n<div id=\"q581351\" class=\"hidden-answer\" style=\"display: none\">Rewrite the expression with the fractional exponent as a radical. The denominator of the fraction determines the root, in this case the cube root.<\/p>\n<p style=\"text-align: center\">[latex]\\sqrt[3]{2x}[\/latex]<\/p>\n<p>The parentheses in [latex]{{\\left( 2x \\right)}^{\\frac{1}{3}}}[\/latex] indicate that the exponent refers to everything within the parentheses.<\/p>\n<h4>Answer<\/h4>\n<p>[latex]{{(2x)}^{^{\\frac{1}{3}}}}=\\sqrt[3]{2x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Remember that exponents only refer to the quantity immediately to their left unless a grouping symbol is used. The example below looks very similar to the previous example with one important difference\u2014there are no parentheses! Look what happens.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Express [latex]2{{x}^{^{\\frac{1}{2}}}}[\/latex] in radical form.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q236347\">Show Solution<\/span><\/p>\n<div id=\"q236347\" class=\"hidden-answer\" style=\"display: none\">Rewrite the expression with the fractional exponent as a radical. The denominator of the fraction determines the root, in this case the cube root.<\/p>\n<p style=\"text-align: center\">[latex]2\\sqrt{x}[\/latex]<\/p>\n<p>The exponent refers only to the part of the expression immediately to the left of the exponent, in this case <i>x, <\/i>but not the 2.<\/p>\n<h4>Answer<\/h4>\n<p>[latex]2{{x}^{^{\\frac{1}{2}}}}=2\\sqrt{x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The next example is intended to help you practice\u00a0placing a rational exponent on the appropriate\u00a0terms in an expression that is written in radical form<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Express [latex]4\\sqrt[3]{xy}[\/latex] with rational exponents.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q527560\">Show Solution<\/span><\/p>\n<div id=\"q527560\" class=\"hidden-answer\" style=\"display: none\">Rewrite the radical using a rational exponent. The root determines the fraction. In this case, the index of the radical is 3, so the rational exponent will be [latex]\\frac{1}{3}[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]4{{(xy)}^{\\frac{1}{3}}}[\/latex]<\/p>\n<p>Since 4 is outside the radical, it is not included in the grouping symbol and the exponent does not refer to it.<\/p>\n<h4>Answer<\/h4>\n<p>[latex]4\\sqrt[3]{xy}=4{{(xy)}^{\\frac{1}{3}}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the next video, we show examples of converting between radical and exponent form.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-7\" title=\"Write Expressions Using Radicals and Rational Exponents\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/5cWkVrANBWA?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>When converting from radical to rational exponent notation, the degree of the root becomes the denominator of the exponent. If you start with a square root, you will have an exponent of [latex]\\frac{1}{2}[\/latex] on the expression in the radical (the radicand). On the other hand, if you start with an exponent of [latex]\\frac{1}{3}[\/latex] you will use a cube root. The following statement summarizes this idea.<\/p>\n<div class=\"textbox shaded\">\n<h3>Writing Fractional Exponents<\/h3>\n<p>Any radical in the form [latex]\\sqrt[n]{a}[\/latex]\u00a0 can be written using a fractional exponent in the form [latex]a^{\\frac{1}{n}}[\/latex].<\/p>\n<\/div>\n<h2>Simplifying Radical Expressions Using Rational Exponents\u00a0and the Laws of Exponents<\/h2>\n<p>Let\u2019s explore some radical expressions now and see how to simplify them. Let&#8217;s start by simplifying this expression,\u00a0\u00a0[latex]\\sqrt[3]{{{a}^{6}}}[\/latex].<\/p>\n<p>One method of simplifying this expression is to factor and pull out groups of [latex]a^{3}[\/latex], as shown below in this example.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\sqrt[3]{{{a}^{6}}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q235013\">Show Solution<\/span><\/p>\n<div id=\"q235013\" class=\"hidden-answer\" style=\"display: none\">Rewrite by factoring out cubes.<\/p>\n<p style=\"text-align: center\">[latex]\\sqrt[3]{{{a}^{3}}\\cdot {{a}^{3}}}[\/latex]<\/p>\n<p>Write each factor under its own radical and simplify.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}\\sqrt[3]{{{a}^{3}}}\\cdot \\sqrt[3]{{{a}^{3}}}\\\\a\\cdot{a}\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\sqrt[3]{{{a}^{6}}}={{a}^{2}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>You can also simplify this expression by thinking about the radical as an expression with a rational exponent, and using the principle that any radical in the form [latex]\\sqrt[n]{{{a}^{x}}}[\/latex] can be written using a fractional exponent in the form [latex]{{a}^{\\tfrac{x}{n}}}[\/latex].<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\sqrt[3]{{{a}^{6}}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q898415\">Show Solution<\/span><\/p>\n<div id=\"q898415\" class=\"hidden-answer\" style=\"display: none\">Rewrite the radical using a rational exponent.<\/p>\n<p style=\"text-align: center\">[latex]{{a}^{\\frac{6}{3}}}[\/latex]<\/p>\n<p>Simplify the exponent.<\/p>\n<p style=\"text-align: center\">[latex]{{a}^{2}}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\sqrt[3]{{{a}^{6}}}={{a}^{2}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Note that rational exponents are subject to all of the same rules as other exponents when they appear in algebraic expressions.<\/p>\n<p>Both simplification methods gave the same result, [latex]a^{2}[\/latex]. Depending on the context of the problem, it may be easier to use one method or the other, but for now, you\u2019ll note that you were able to simplify this expression more quickly using rational exponents than when using the \u201cpull-out\u201d method.<\/p>\n<p>Let\u2019s try another example.<\/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\">Rewrite the radical using rational exponents.<\/p>\n<p style=\"text-align: center\">[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 style=\"text-align: center\">[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 style=\"text-align: center\">[latex]3{{x}^{2}}\\sqrt[4]{{{y}^{3}}}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\sqrt[4]{81{{x}^{8}}{{y}^{3}}}=3{{x}^{2}}\\sqrt[4]{{{y}^{3}}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Again, the alternative method is to work on simplifying under the radical by using factoring. For the example you just solved, it looks like this.<\/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\">Rewrite the expression.<\/p>\n<p style=\"text-align: center\">[latex]\\sqrt[4]{81}\\cdot \\sqrt[4]{{{x}^{8}}}\\cdot \\sqrt[4]{{{y}^{3}}}[\/latex]<\/p>\n<p>Factor each radicand.<\/p>\n<p style=\"text-align: center\">[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 style=\"text-align: center\">[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<h4>Answer<\/h4>\n<p>[latex]\\sqrt[4]{81x^{8}y^{3}}=3x^{2}\\sqrt[4]{y^{3}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The following video shows more examples of how to simplify a radical expression using rational exponents.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-8\" 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\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-4502\">\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 (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\/B3riJsl7uZM\">https:\/\/youtu.be\/B3riJsl7uZM<\/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 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>Simplify Cube Roots (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\/9Nh-Ggd2VJo\">https:\/\/youtu.be\/9Nh-Ggd2VJo<\/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=\"http:\/\/Simplify%20Cube%20Roots%20(Not%20Perfect%20Cube%20Radicands)\">http:\/\/Simplify%20Cube%20Roots%20(Not%20Perfect%20Cube%20Radicands)<\/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 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>Write Basic Expression in Radical Form and 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\/eGJgmo2CpN4\">https:\/\/youtu.be\/eGJgmo2CpN4<\/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>Write Expressions Using Radicals and Rational Exponents. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/5cWkVrANBWA\">https:\/\/youtu.be\/5cWkVrANBWA<\/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><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Unit 16: Radical Expressions and Quadratic Equations, from Developmental Math: An Open Program. <strong>Provided by<\/strong>: Monterey Institute of Technology and Education. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/\">http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/<\/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":25777,"menu_order":4,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Unit 16: Radical Expressions and Quadratic Equations, from Developmental Math: An Open Program\",\"author\":\"\",\"organization\":\"Monterey Institute of Technology and 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