{"id":1646,"date":"2016-06-22T13:19:16","date_gmt":"2016-06-22T13:19:16","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/?post_type=chapter&#038;p=1646"},"modified":"2016-10-03T21:24:20","modified_gmt":"2016-10-03T21:24:20","slug":"1646","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/tallahassee-intermediatealgebra\/chapter\/1646\/","title":{"raw":"Simplify Expressions with Roots and Rational Exponents","rendered":"Simplify Expressions with Roots and Rational Exponents"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Introduction to Roots\r\n<ul>\r\n \t<li>Define and evaluate principal square roots<\/li>\r\n \t<li>Define and evaluate nth roots<\/li>\r\n \t<li>Estimate roots that are not perfect<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Radical Expressions and Rational Exponents\r\n<ul>\r\n \t<li>Define and identify a radical expression<\/li>\r\n \t<li>Convert radicals to expressions with rational exponents<\/li>\r\n \t<li>Convert expressions with rational exponents to their radical equivalent<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Simplify Radical Expressions\r\n<ul>\r\n \t<li>Simplify radical expressions using factoring<\/li>\r\n \t<li>Simplify radical expressions\u00a0using rational exponents\u00a0and the laws of exponents<\/li>\r\n \t<li>Define [latex]\\sqrt{x^2}=|x|[\/latex], and apply it when simplifying radical expressions<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\nDid you know that you can take the 6th root of a number? You have probably heard of a square root, written [latex]\\sqrt{}[\/latex], but you can also take a third, fourth and even a 5,000th root (if you really had to). In this lesson we will learn how a square root is defined and then we will build on that to form an understanding of nth roots. \u00a0We will use factoring and rules for exponents to simplify mathematical expressions that contain roots.\r\n\r\nThe most common root is the <strong>square root<\/strong>. First, we will define what square roots are,\u00a0 and how you find the square root of a number. Then we will apply similar ideas to define and evaluate nth roots.\r\n\r\nRoots are the inverse of exponents, much like multiplication is the inverse of division. Recall\u00a0how exponents are defined, and written; with an exponent, as words, and as repeated multiplication.\r\n\r\n<strong>Exponent:<\/strong> [latex] {{3}^{2}}[\/latex],\u00a0[latex] {{4}^{5}}[\/latex],\u00a0[latex] {{x}^{3}}[\/latex],\u00a0[latex] {{x}^{\\text{n}}}[\/latex]\r\n\r\n<strong>Name:<\/strong>\u00a0\u201cThree squared\u201d or\u00a0\u201cThree to the second power\u201d,\u00a0\u201cFour to the fifth power\u201d,\u00a0\u201c<i>x<\/i> cubed\u201d,\u00a0\u201c<i>x<\/i> to the <i>n<\/i>th power\u201d\r\n\r\n<strong>Repeated Multiplication:<\/strong>\u00a0[latex] 3\\cdot 3[\/latex], \u00a0[latex] 4\\cdot 4\\cdot 4\\cdot 4\\cdot 4[\/latex], \u00a0[latex] x\\cdot x\\cdot x[\/latex], \u00a0[latex] \\underbrace{x\\cdot x\\cdot x...\\cdot x}_{n\\text{ times}}[\/latex].\r\n\r\nConversely,\u00a0 when you are trying to find the square root of a number (say, 25), you are trying to find a number that can be multiplied by itself to create that original number. In the case of 25, you can find that [latex]5\\cdot5=25[\/latex], so 5 must be the square root.\r\n<h2>Square Roots<\/h2>\r\nThe symbol for the square root is called a <strong>radical symbol<\/strong> and looks like this: [latex]\\sqrt{\\,\\,\\,}[\/latex]. The expression [latex] \\sqrt{25}[\/latex] is read \u201cthe square root of twenty-five\u201d or \u201cradical twenty-five.\u201d The number that is written under the radical symbol is called the <strong>radicand<\/strong>.\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200220\/CNX_CAT_Figure_01_03_002.jpg\" alt=\"The expression: square root of twenty-five is enclosed in a circle. The circle has an arrow pointing to it labeled: Radical expression. The square root symbol has an arrow pointing to it labeled: Radical. The number twenty-five has an arrow pointing to it labeled: Radicand.\" data-media-type=\"image\/jpg\" \/>\r\n\r\nThe following table shows different radicals and their equivalent written and simplified forms.\r\n<table style=\"width: 70%;\">\r\n<thead>\r\n<tr>\r\n<th>Radical<\/th>\r\n<th>Name<\/th>\r\n<th>Simplified Form<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex] \\sqrt{36}[\/latex]<\/td>\r\n<td>\n\n\u201cSquare root of thirty-six\u201d\r\n\r\n\u201cRadical thirty-six\u201d<\/td>\r\n<td>[latex] \\sqrt{36}=\\sqrt{6\\cdot 6}=6[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex] \\sqrt{100}[\/latex]<\/td>\r\n<td>\n\n\u201cSquare root of one hundred\u201d\r\n\r\n\u201cRadical one hundred\u201d<\/td>\r\n<td>[latex] \\sqrt{100}=\\sqrt{10\\cdot 10}=10[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex] \\sqrt{225}[\/latex]<\/td>\r\n<td>\n\n\u201cSquare root of two hundred twenty-five\u201d\r\n\r\n\u201cRadical two hundred twenty-five\u201d<\/td>\r\n<td>[latex] \\sqrt{225}=\\sqrt{15\\cdot 15}=15[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nConsider [latex] \\sqrt{25}[\/latex] again. You may realize that there is another value that, when multiplied by itself, also results in 25. That number is [latex]\u22125[\/latex].\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{r}5\\cdot 5=25\\\\-5\\cdot -5=25\\end{array}[\/latex]<\/p>\r\nBy definition, the square root symbol always means to find the positive root, called the <strong>principal root<\/strong>. So while [latex]5\\cdot5[\/latex] and [latex]\u22125\\cdot\u22125[\/latex] both equal 25, only 5 is the principal root. You should also know that zero is special because it has only one square root: itself (since [latex]0\\cdot0=0[\/latex]).\r\n\r\nIn our first example we will show you how to use radical notation to evaluate principal square roots.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nFind the principal root of each expression.\r\n<ol>\r\n \t<li>[latex]\\sqrt{100}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{16}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{25+144}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{49}-\\sqrt{81}\\\\[\/latex]<\/li>\r\n \t<li>[latex] -\\sqrt{81}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{-9}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"419579\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"419579\"]\r\n<ol>\r\n \t<li>[latex]\\sqrt{100}=10[\/latex] because [latex]{10}^{2}=100[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{\\sqrt{16}}=\\sqrt{4}=2[\/latex] because [latex]{4}^{2}=16[\/latex] and [latex]{2}^{2}=4[\/latex]<\/li>\r\n \t<li>Recall that square roots act as grouping symbols in the order of operations, so addition and subtraction must be performed first when they occur under a radical. [latex]\\sqrt{25+144}=\\sqrt{169}=13[\/latex] because [latex]{13}^{2}=169[\/latex]<\/li>\r\n \t<li>This problem is similar to the last one, but this time subtraction should occur after evaluating the root. Stop and think about why these two problems are different. [latex]\\sqrt{49}-\\sqrt{81}=7 - 9=-2[\/latex] because [latex]{7}^{2}=49[\/latex] and [latex]{9}^{2}=81[\/latex]<\/li>\r\n \t<li>\r\n<p style=\"text-align: left;\">The negative in front means to take the opposite of the value after you simplify the radical. [latex] -\\sqrt{81}\\\\-\\sqrt{9\\cdot 9}[\/latex].\u00a0 The square root of 81 is 9. Then, take the opposite of 9. [latex]\u2212(9)[\/latex]<\/p>\r\n<\/li>\r\n \t<li>[latex]\\sqrt{-9}[\/latex], we are looking for a number that when it is squared, returns [latex]-9[\/latex]. We can try [latex](-3)^2[\/latex], but that will give a positive result, and [latex]3^2[\/latex] will also give a positive result. This leads to an important fact - \u00a0you cannot find the square root of a negative number.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video we present more examples of how to find a principle square root.\r\n\r\nhttps:\/\/youtu.be\/2cWAkmJoaDQ\r\n\r\nThe last example we showed leads to an important characteristic of square roots. You can only take the square root of values that are nonnegative.\r\n<p class=\"textbox shaded\"><strong>Domain of a Square Root<\/strong>\r\n[latex]\\sqrt{-a}[\/latex] is not defined for all real numbers, a. Therefore, [latex]\\sqrt{a}[\/latex] is defined for [latex]a\\ge0[\/latex]<\/p>\r\n\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Think About It<\/h3>\r\nDoes [latex]\\sqrt{25}=\\pm 5[\/latex]? Write your ideas and a sentence to defend them in the box below before you look at the answer.\r\n\r\n[practice-area rows=\"1\"][\/practice-area]\r\n[reveal-answer q=\"101071\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"101071\"]\r\n\r\n<em>No. Although both<\/em> [latex]{5}^{2}[\/latex] <em>and<\/em> [latex]{\\left(-5\\right)}^{2}[\/latex] <em>are<\/em> [latex]25[\/latex], <em>the radical symbol implies only a nonnegative root, the principal square root. The principal square root of 25 is<\/em> [latex]\\sqrt{25}=5[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Cube Roots<\/h2>\r\nWe know that [latex]5^2=25, \\text{ and }\\sqrt{25}=5[\/latex] but what if we want to \"undo\" [latex]5^3=125, \\text{ or }5^4=625[\/latex]? We can use higher order roots to answer these questions.\r\n\r\nSuppose we know that [latex]{a}^{3}=8[\/latex]. We want to find what number raised to the 3rd power is equal to 8. Since [latex]{2}^{3}=8[\/latex], we say that 2 is the cube root of 8. In the next example we will evaluate the cube roots of some perfect cubes.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nEvaluate the following:\r\n<ol>\r\n \t<li>[latex] \\sqrt[3]{125}[\/latex]<\/li>\r\n \t<li>[latex] \\sqrt[3]{-8}[\/latex]<\/li>\r\n \t<li>[latex] \\sqrt[3]{27}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"517592\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"517592\"]\r\n\r\n1. You can read this as \u201cthe third root of 125\u201d or \u201cthe cube root of 125.\u201d To evaluate this expression, look for a number that, when multiplied by itself two times (for a total of three identical factors), equals 125. [latex]\\text{?}\\cdot\\text{?}\\cdot\\text{?}=125[\/latex]. Since 125 ends in 5, 5 is a good candidate. [latex]5\/cdot5\/cdot5=125[\/latex]\r\n2. We want to find a number whose cube is 8. [latex]2\\cdot2\\cdot2=8[\/latex] the cube root of 8 is 2.\r\n\r\n3. 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.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nAs we saw in the last example,there is one interesting fact about cube roots that is not true of square roots. Negative numbers can\u2019t have real number square roots, but negative numbers can have real number cube roots! What is the cube root of [latex]\u22128[\/latex]? [latex] \\sqrt[3]{-8}=-2[\/latex] because [latex] -2\\cdot -2\\cdot -2=-8[\/latex]. Remember, when you are multiplying an odd number of negative numbers, the result is negative! Consider [latex] \\sqrt[3]{{{(-1)}^{3}}}=-1[\/latex].\r\n\r\nIn the following video we show more examples of finding a cube root.\r\n\r\nhttps:\/\/youtu.be\/9Nh-Ggd2VJo\r\n<h2>Nth Roots<\/h2>\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\r\nWe can apply the same idea to any exponent and it's corresponding root.\u00a0 The <em>n<\/em>th root of [latex]a[\/latex] is a number that, when raised to the <em>n<\/em>th power, gives [latex]a[\/latex]. For example, [latex]3[\/latex] is the 5th root of [latex]243[\/latex] because [latex]{\\left(3\\right)}^{5}=243[\/latex]. If [latex]a[\/latex] is a real number with at least one <em>n<\/em>th root, then the <strong>principal <em>n<\/em>th root<\/strong> of [latex]a[\/latex] is the number with the same sign as [latex]a[\/latex] that, when raised to the <em>n<\/em>th power, equals [latex]a[\/latex].\r\n\r\nThe principal <em>n<\/em>th root of [latex]a[\/latex] is written as [latex]\\sqrt[n]{a}[\/latex], where [latex]n[\/latex] is a positive integer greater than or equal to 2. In the radical expression, [latex]n[\/latex] is called the <strong>index<\/strong> of the radical.\r\n<div class=\"textbox\">\r\n<h3>Definition:\u00a0Principal <em>n<\/em>th Root<\/h3>\r\nIf [latex]a[\/latex] is a real number with at least one <em>n<\/em>th root, then the <strong>principal <em>n<\/em>th root<\/strong> of [latex]a[\/latex], written as [latex]\\sqrt[n]{a}[\/latex], is the number with the same sign as [latex]a[\/latex] that, when raised to the <em>n<\/em>th power, equals [latex]a[\/latex]. The <strong>index<\/strong> of the radical is [latex]n[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nEvaluate each of the following:\r\n<ol>\r\n \t<li>[latex]\\sqrt[5]{-32}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt[4]{81}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt[8]{-1}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"140298\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"140298\"]\r\n<ol>\r\n \t<li>[latex]\\sqrt[5]{-32}[\/latex] Factor 32, because [latex]{\\left(-2\\right)}^{5}=-32 \\\\ \\text{ }[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt[4]{81}[\/latex]. Factoring can help, we know that [latex]9\\cdot9=81[\/latex] and we can further factor each 9: [latex]\\sqrt[4]{81}=\\sqrt[4]{3\\cdot3\\cdot3\\cdot3}=\\sqrt[4]{3^4}=3[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt[8]{-1}[\/latex], since we have an 8th root - which is even- with a negative number as the radicand, this root has no real number solutions. In other words, [latex]-1\\cdot-1\\cdot-1\\cdot-1\\cdot-1\\cdot-1\\cdot-1\\cdot-1=+1[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video we show more examples of how to evaluate and nth root.\r\n\r\nhttps:\/\/youtu.be\/vA2DkcUSRSk\r\n\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 evaluate the radicals [latex] \\sqrt[3]{-81},\\ \\sqrt[5]{-64}[\/latex], and [latex] \\sqrt[7]{-2187}[\/latex], but you cannot evaluate the radicals [latex] \\sqrt[{}]{-100},\\ \\sqrt[4]{-16}[\/latex], or [latex] \\sqrt[6]{-2,500}[\/latex].\r\n<h2>Estimate Roots<\/h2>\r\nAn approach to handling roots that are not perfect (squares, cubes, etc.)\u00a0 is to approximate them by comparing the values to perfect squares, cubes, or nth roots. Suppose you wanted to know the square root of 17. Let\u2019s look at how you might approximate it.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nEstimate. [latex] \\sqrt{17}[\/latex]\r\n\r\n[reveal-answer q=\"358591\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"358591\"]Think of two perfect squares that surround 17.\u00a017 is in between the perfect squares 16 and 25.\u00a0So, [latex] \\sqrt{17}[\/latex] must be in between [latex] \\sqrt{16}[\/latex] and [latex] \\sqrt{25}[\/latex].\r\n\r\nDetermine whether [latex] \\sqrt{17}[\/latex] is closer to 4 or to 5 and make another estimate.\r\n<p style=\"text-align: center;\">[latex] \\sqrt{16}=4[\/latex] and [latex] \\sqrt{25}=5[\/latex]<\/p>\r\nSince 17 is closer to 16 than 25, [latex] \\sqrt{17}[\/latex] is probably about 4.1 or 4.2.\r\n\r\nUse trial and error to get a better estimate of [latex] \\sqrt{17}[\/latex]. Try squaring incrementally greater numbers, beginning with 4.1, to find a good approximation for [latex] \\sqrt{17}[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\left(4.1\\right)^{2}[\/latex]<\/p>\r\n[latex]\\left(4.1\\right)^{2}[\/latex]\u00a0gives a closer estimate than [latex](4.2)^{2}[\/latex].\r\n<p style=\"text-align: center;\">[latex]4.1\\cdot4.1=16.81\\\\4.2\\cdot4.2=17.64[\/latex]<\/p>\r\nContinue to use trial and error to get an even better estimate.\r\n<p style=\"text-align: center;\">[latex]4.12\\cdot4.12=16.9744\\\\4.13\\cdot4.13=17.0569[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex] \\sqrt{17}\\approx 4.12[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThis approximation is pretty close. If you kept using this trial and error strategy you could continue to find the square root to the thousandths, ten-thousandths, and hundred-thousandths places, but eventually it would become too tedious to do by hand.\r\n\r\nFor this reason, when you need to find a more precise approximation of a square root, you should use a calculator. Most calculators have a square root key [latex] (\\sqrt{{}})[\/latex] that will give you the square root approximation quickly. On a simple 4-function calculator, you would likely key in the number that you want to take the square root of and then press the square root key.\r\n\r\nTry to find [latex] \\sqrt{17}[\/latex] using your calculator. Note that you will not be able to get an \u201cexact\u201d answer because [latex] \\sqrt{17}[\/latex] is an irrational number, a number that cannot be expressed as a fraction, and the decimal never terminates or repeats. To nine decimal positions, [latex] \\sqrt{17}[\/latex] is approximated as 4.123105626. A calculator can save a lot of time and yield a more precise square root when you are dealing with numbers that aren\u2019t perfect squares.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nApproximate [latex] \\sqrt[3]{30}[\/latex] and also find its value using a calculator.\r\n\r\n[reveal-answer q=\"71092\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"71092\"]Find the cubes that surround 30.\r\n\r\n30 is inbetween the perfect cubes 27 and 81.\r\n\r\n[latex] \\sqrt[3]{27}=3[\/latex] and [latex] \\sqrt[3]{81}=4[\/latex], so [latex] \\sqrt[3]{30}[\/latex] is between 3 and 4.\r\nUse a calculator.\r\n<p style=\"text-align: center;\">[latex]\\sqrt[3]{30}\\approx3.10723[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\nBy approximation: [latex]3\\ge\\sqrt[3]{30}\\le4[\/latex]\r\n\r\nUsing a calculator: [latex] \\sqrt[3]{30}\\approx3.10723[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe following video shows another example of how to estimate a square root.\r\n\r\nhttps:\/\/youtu.be\/iNfalyW7olk\r\n<h2>Radical Expressions and Rational Exponents<\/h2>\r\nSquare roots are most often written using a radical sign, like this, [latex] \\sqrt{4}[\/latex]. But there is another way to represent them. You can use rational exponents instead of a radical. A <strong>rational exponent<\/strong> is an exponent that is a fraction. For example, [latex] \\sqrt{4}[\/latex] can be written as [latex] {{4}^{\\tfrac{1}{2}}}[\/latex].\r\n\r\nCan\u2019t imagine raising a number to a rational exponent? They may be hard to get used to, but rational exponents can actually help simplify some problems. Writing radicals with rational exponents will come in handy when we discuss techniques for simplifying more complex radical expressions.\r\n\r\n<strong>Radical expressions<\/strong> are expressions that contain radicals. Radical expressions come in many forms, from simple and familiar, such as[latex] \\sqrt{16}[\/latex], to quite complicated, as in [latex] \\sqrt[3]{250{{x}^{4}}y}[\/latex]\r\n<h2>Write an expression with a rational exponent as a radical<\/h2>\r\nRadicals and fractional exponents are alternate ways of expressing the same thing. \u00a0In the table below we show equivalent ways to express radicals: with a root, with a rational exponent, and as a principal root.\r\n<table style=\"width: 30%;\">\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;\">Principal Root<\/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{16}[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex] {{16}^{\\tfrac{1}{2}}}[\/latex]<\/td>\r\n<td style=\"text-align: center;\">4<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">[latex] \\sqrt{25}[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex] {{25}^{\\tfrac{1}{2}}}[\/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{100}[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex] {{100}^{\\tfrac{1}{2}}}[\/latex]<\/td>\r\n<td style=\"text-align: center;\">10<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nLet\u2019s look at some more examples, but this time with cube roots. Remember, cubing a number raises it to the power of three. Notice that in the examples in the table below, the denominator of the rational exponent is the number 3.\r\n<table style=\"width: 30%;\">\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;\">Principal Root<\/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: 30%;\">\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\nIn 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]{{}}[\/latex], and [latex] \\frac{1}{8}[\/latex] translates to the eighth root or [latex] \\sqrt[8]{{}}[\/latex].\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}{3}}}}[\/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[3]{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}{3}}}}=2\\sqrt[3]{x}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Write a radical expression as an expression with a rational exponent<\/h2>\r\n[caption id=\"attachment_3123\" align=\"alignright\" width=\"141\"]<img class=\" wp-image-3123\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/26174543\/Screen-Shot-2016-07-26-at-10.44.01-AM-300x291.png\" alt=\"Person sitting on the ground with one leg arched behind them and one leg curved in front of them.\" width=\"141\" height=\"137\" \/> Flexibility[\/caption]\r\n\r\nWe can write radicals with rational exponents, and as we will see when we simplify more complex radical expressions, this can make things easier. Having different ways to express and write algebraic expressions allows us to have flexibility in solving and simplifying them. It is like having a thesaurus when you write, you want to have options for expressing yourself!\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nWrite [latex] \\sqrt[4]{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] \\Large\\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\r\n[latex] {{81}^{\\frac{1}{4}}}[\/latex]\r\n<h4>Answer<\/h4>\r\n[latex] \\sqrt[4]{81}={{81}^{\\frac{1}{4}}}[\/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] 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\n<h2>Rational exponents whose numerator is not equal to one<\/h2>\r\nAll of the numerators for the fractional exponents in the examples above were 1. You can use fractional exponents that have numerators other than 1 to express roots, as shown below.\r\n<table>\r\n<thead>\r\n<tr>\r\n<th>\r\n<p style=\"text-align: center;\">Radical<\/p>\r\n<\/th>\r\n<th>\r\n<p style=\"text-align: center;\">Exponent<\/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{9}[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]9^{\\frac{1}{2}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">[latex] \\sqrt[3]{{{9}^{2}}}[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]9^{\\frac{2}{3}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">[latex]\\sqrt[4]{9^{3}}[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]9^{\\frac{3}{4}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">[latex]\\sqrt[5]{9^{2}}[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]9^{\\frac{2}{5}}[\/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]{9^{x}}[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]9\\frac{x}{n}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<img class=\" wp-image-3198 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/29225734\/Screen-Shot-2016-07-29-at-3.56.45-PM-300x179.png\" alt=\"Screen Shot 2016-07-29 at 3.56.45 PM\" width=\"380\" height=\"227\" \/>\r\n\r\nTo rewrite a radical using a fractional exponent, the power to which the radicand is raised becomes the numerator and the root\/ index becomes the denominator.\r\n<div class=\"textbox shaded\">\r\n<h3>Writing Rational\u00a0Exponents<\/h3>\r\nAny radical in the form [latex]\\sqrt[n]{a^{x}}[\/latex]\u00a0 can be written using a fractional exponent in the form [latex]a^{\\frac{x}{n}}[\/latex].\r\n\r\n<\/div>\r\nThe relationship between [latex] \\sqrt[n]{{{a}^{x}}}[\/latex]and [latex] {{a}^{\\frac{x}{n}}}[\/latex] works for rational exponents that have a numerator of 1 as well. For example, the radical [latex] \\sqrt[3]{8}[\/latex] can also be written as [latex] \\sqrt[3]{{{8}^{1}}}[\/latex], since any number remains the same value if it is raised to the first power. You can now see where the numerator of 1 comes from in the equivalent form of [latex] {{8}^{\\frac{1}{3}}}[\/latex].\r\n\r\nIn the next example, we practice writing radicals with rational exponents where the numerator is not equal to one.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nRewrite the radicals using a rational exponent, then simplify your result.\r\n<ol>\r\n \t<li>[latex]\\sqrt[3]{{{a}^{6}}}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt[12]{16^3}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"898415\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"898415\"]\r\n\r\n1.[latex]\\sqrt[n]{a^{x}}[\/latex] can be rewritten as\u00a0[latex]a^{\\frac{x}{n}}[\/latex], so in this case [latex]n=3,\\text{ and }x=6[\/latex], therefore\r\n\r\n[latex]\\sqrt[3]{{{a}^{6}}}={{a}^{\\frac{6}{3}}}[\/latex]\r\n\r\nSimplify the exponent.\r\n\r\n[latex]{{a}^{\\frac{6}{3}}}={{a}^{2}}[\/latex]\r\n<h4>Answer<\/h4>\r\n[latex] \\sqrt[3]{{{a}^{6}}}={{a}^{2}}[\/latex]\r\n\r\n2.\u00a0[latex]\\sqrt[n]{a^{x}}[\/latex] can be rewritten as\u00a0[latex]a^{\\frac{x}{n}}[\/latex], so in this case [latex]n=12,\\text{ and }x=3[\/latex], therefore\r\n<p style=\"text-align: center;\">[latex]\\sqrt[12]{16^3}={16}^{\\frac{3}{12}}={16}^{\\frac{1}{4}}[\/latex]<\/p>\r\nSimplify the expression using rules for exponents.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{ccc}16=2^4\\\\{16}^{\\frac{1}{4}}={2^4}^{\\frac{1}{4}}\\\\=2^{4\\cdot\\frac{1}{4}}\\\\=2^1=2\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\sqrt[12]{16^3}=2[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn our last example we will rewrite expressions with rational exponents as radicals. This practice will help us when we simplify more complicated radical expressions, and as we learn how to solve radical equations. Typically it is easier to simplify when we use rational exponents, but this exercise is intended to help you understand how\u00a0the numerator and denominator of the exponent are\u00a0the exponent of a radicand and index of a radical.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nRewrite the expressions\u00a0using a radical.\r\n<ol>\r\n \t<li>[latex]{x}^{\\frac{2}{3}}[\/latex]<\/li>\r\n \t<li>[latex]{5}^{\\frac{4}{7}}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"200228\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"200228\"]\r\n<ol>\r\n \t<li>[latex]{x}^{\\frac{2}{3}}[\/latex], the numerator is 2 and the denominator is 3, therefore we will have the third root of x squared, [latex]\\sqrt[3]{x^2}[\/latex]<\/li>\r\n \t<li>[latex]{5}^{\\frac{4}{7}}[\/latex], the numerator is 4 and the denominator is 7, so we will have the seventh root of 5 raised to the fourth power. [latex]\\sqrt[7]{5^4}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video we show more examples of writing radical expressions with rational exponents and expressions with rational exponents as radical expressions.\r\n\r\nhttps:\/\/youtu.be\/5cWkVrANBWA\r\n\r\nWe will use this notation later, so come back for practice if you forget how\u00a0to write a radical with a rational exponent.\r\n<h2>Simplify Radical Expressions<\/h2>\r\n<strong>Radical expressions<\/strong> are expressions that contain radicals. Radical expressions come in many forms, from simple and familiar, such as[latex] \\sqrt{16}[\/latex], to quite complicated, as in [latex] \\sqrt[3]{250{{x}^{4}}y}[\/latex].\r\n\r\nTo simplify complicated radical expressions, we can use some definitions and rules from simplifying exponents. Recall the <strong>Product Raised to a Power Rule<\/strong> from when you studied exponents. This rule states that the product of two or more non-zero numbers raised to a power is equal to the product of each number raised to the same power. In math terms, it is written [latex]\\left(ab\\right)^{x}=a^{x}\\cdot{b}^{x}[\/latex].\u00a0So, for example, you can use the rule to rewrite [latex] {{\\left( 3x \\right)}^{2}}[\/latex] as [latex] {{3}^{2}}\\cdot {{x}^{2}}=9\\cdot {{x}^{2}}=9{{x}^{2}}[\/latex].\r\n\r\nNow instead of using the exponent 2, let\u2019s use the exponent [latex] \\frac{1}{2}[\/latex]. The exponent is distributed in the same way.\r\n<p style=\"text-align: center;\">[latex] {{\\left( 3x \\right)}^{\\frac{1}{2}}}={{3}^{\\frac{1}{2}}}\\cdot {{x}^{\\frac{1}{2}}}[\/latex]<\/p>\r\nAnd since you know that raising a number to the [latex] \\frac{1}{2}[\/latex] power is the same as taking the square root of that number, you can also write it this way.\r\n<p style=\"text-align: center;\">[latex] \\sqrt{3x}=\\sqrt{3}\\cdot \\sqrt{x}[\/latex]<\/p>\r\nLook at that\u2014you can think of any number underneath a radical as the <i>product of separate factors<\/i>, each underneath its own radical.\r\n<div class=\"textbox shaded\">\r\n<h3>A Product Raised to a Power Rule\u00a0or sometimes called\u00a0The Square Root of a Product Rule<\/h3>\r\nFor any real numbers <i>a<\/i> and <i>b<\/i>, [latex] \\sqrt{ab}=\\sqrt{a}\\cdot \\sqrt{b}[\/latex].\r\n\r\nFor example: [latex] \\sqrt{100}=\\sqrt{10}\\cdot \\sqrt{10}[\/latex], and [latex] \\sqrt{75}=\\sqrt{25}\\cdot \\sqrt{3}[\/latex]\r\n\r\n<\/div>\r\nThis rule is important because it helps you think of one radical as the product of multiple radicals. If you can identify perfect squares within a radical, as with [latex] \\sqrt{(2\\cdot 2)(2\\cdot 2)(3\\cdot 3})[\/latex], you can rewrite the expression as the product of multiple perfect squares: [latex] \\sqrt{{{2}^{2}}}\\cdot \\sqrt{{{2}^{2}}}\\cdot \\sqrt{{{3}^{2}}}[\/latex].\r\n<p class=\"p1\">The square root of a product rule will help us simplify roots that aren't perfect, as is shown the following example.<\/p>\r\n\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\sqrt{63}[\/latex]\r\n[reveal-answer q=\"908978\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"908978\"]63 is not a perfect square so we can use the\u00a0square root of a product rule to simplify any factors that are perfect squares.\r\nFactor 63 into 7 and 9.\r\n[latex] \\sqrt{7\\cdot 9}[\/latex]\r\n9 is a perfect square, [latex]9=3^2[\/latex], therefore we can rewrite the radicand.\r\n\r\n[latex] \\sqrt{7\\cdot {{3}^{2}}}[\/latex]\r\n\r\nUsing the Product Raised to a Power rule, separate the radical into the product of two factors, each under a radical.\r\n[latex] \\sqrt{7}\\cdot \\sqrt{{{3}^{2}}}[\/latex]\r\nTake the square root of [latex]3^{2}[\/latex].\r\n[latex] \\sqrt{7}\\cdot 3[\/latex]\r\nRearrange factors so the integer appears before the radical, and then multiply. (This is done so that it is clear that only the 7 is under the radical, not the 3.)\r\n[latex] 3\\cdot \\sqrt{7}[\/latex]\r\n<b>Answer<\/b>\r\n[latex] \\sqrt{63}=3\\sqrt{7}[\/latex]\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe final answer [latex] 3\\sqrt{7}[\/latex] may look a bit odd, but it is in simplified form. You can read this as \u201cthree radical seven\u201d or \u201cthree times the square root of seven.\u201d\r\n\r\nThe following video shows more examples of how to simplify square roots that do not have perfect square radicands.\r\n\r\nhttps:\/\/youtu.be\/oRd7aBCsmfU\r\n\r\nBefore we move on to simplifying more complex radicals with variables, we need to learn about an important behavior of square roots with variables in the radicand.\r\n\r\nConsider the expression [latex] \\sqrt{{{x}^{2}}}[\/latex]. This looks like it should be equal to <i>x<\/i>, right? 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 style=\"width: 40%;\">\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]! However, in all cases [latex]\\sqrt{x^{2}}=\\left|x\\right|[\/latex].\u00a0You need to consider this fact when simplifying radicals with an even index that contain variables, because by definition [latex]\\sqrt{x^{2}}[\/latex]\u00a0is always nonnegative.\r\n<div class=\"textbox shaded\">\r\n<h3>Taking the Square Root of a Radical Expression<\/h3>\r\nWhen finding the square root of an expression that contains variables raised to 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\nWe will combine this with the square root of a product rule in our next example to simplify an expression with three variables in the radicand.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\sqrt{{{a}^{3}}{{b}^{5}}{{c}^{2}}}[\/latex]\r\n\r\n[reveal-answer q=\"141094\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"141094\"]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.\r\n\r\n[latex] \\left| ac \\right|{{b}^{2}}\\sqrt{ab}[\/latex]\r\n<h4>Answer<\/h4>\r\n[latex] \\sqrt{{{a}^{3}}{{b}^{5}}{{c}^{2}}}=\\left| ac \\right|{{b}^{2}}\\sqrt{ab}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3>Analysis of the Solution<\/h3>\r\nWhy didn't we write [latex]b^2[\/latex] as [latex]|b^2|[\/latex]? \u00a0Because when you square a number, you will always get a positive result, so the principal square root of\u00a0[latex]\\left(b^2\\right)^2[\/latex] will always be non-negative. One tip for\u00a0knowing when to apply the absolute value after simplifying any even indexed root is to look at the final exponent on your variable terms. \u00a0If the exponent is odd - including 1 - add an absolute value. This applies to simplifying any root with an even index, as we will see in later examples.\r\n\r\nIn the following video you will see more examples of how to simplify radical expressions with variables.\r\n\r\nhttps:\/\/youtu.be\/q7LqsKPoAKo\r\n\r\nWe will show another example where the simplified expression contains variables with both odd and even powers.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\sqrt{9{{x}^{6}}{{y}^{4}}}[\/latex]\r\n\r\n[reveal-answer q=\"41297\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"41297\"]Factor to find identical pairs.\r\n<p style=\"text-align: center;\">[latex] \\sqrt{3\\cdot 3\\cdot {{x}^{3}}\\cdot {{x}^{3}}\\cdot {{y}^{2}}\\cdot {{y}^{2}}}[\/latex]<\/p>\r\nRewrite the pairs as perfect squares.\r\n<p style=\"text-align: center;\">[latex] \\sqrt{{{3}^{2}}\\cdot {{\\left( {{x}^{3}} \\right)}^{2}}\\cdot {{\\left( {{y}^{2}} \\right)}^{2}}}[\/latex]<\/p>\r\nSeparate into individual radicals.\r\n<p style=\"text-align: center;\">[latex] \\sqrt{{{3}^{2}}}\\cdot \\sqrt{{{\\left( {{x}^{3}} \\right)}^{2}}}\\cdot \\sqrt{{{\\left( {{y}^{2}} \\right)}^{2}}}[\/latex]<\/p>\r\nSimplify.\r\n<p style=\"text-align: center;\">[latex] 3{{x}^{3}}{{y}^{2}}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">Because x has an odd power, we will add the absolute value for our final solution.<\/p>\r\n<p style=\"text-align: center;\">[latex] 3|{{x}^{3}}|{{y}^{2}}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex] \\sqrt{9{{x}^{6}}{{y}^{4}}}=3|{{x}^{3}}|{y}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn our next example we will start with an expression written with a rational exponent. You will see that you can use a similar process - factoring and sorting terms into squares - to simplify this expression.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] {{(36{{x}^{4}})}^{\\frac{1}{2}}}[\/latex]\r\n\r\n[reveal-answer q=\"554375\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"554375\"]Rewrite the expression with the fractional exponent as a radical.\r\n<p style=\"text-align: center;\">[latex] \\sqrt{36{{x}^{4}}}[\/latex]<\/p>\r\nFind the square root of both the coefficient and the variable.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r} \\sqrt{{{6}^{2}}\\cdot {{x}^{4}}}\\\\\\sqrt{{{6}^{2}}}\\cdot \\sqrt{{{x}^{4}}}\\\\\\sqrt{{{6}^{2}}}\\cdot \\sqrt{{{({{x}^{2}})}^{2}}}\\\\6\\cdot{x}^{2}\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex] {{(36{{x}^{4}})}^{\\frac{1}{2}}}=6{{x}^{2}}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nHere is one more example with perfect squares.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\sqrt{49{{x}^{10}}{{y}^{8}}}[\/latex]\r\n\r\n[reveal-answer q=\"283065\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"283065\"]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<p style=\"text-align: center;\">[latex] \\sqrt{7\\cdot 7\\cdot {{x}^{5}}\\cdot {{x}^{5}}\\cdot {{y}^{4}}\\cdot {{y}^{4}}}[\/latex]<\/p>\r\nRewrite the pairs as squares.\r\n<p style=\"text-align: center;\">[latex] \\sqrt{{{7}^{2}}\\cdot {{({{x}^{5}})}^{2}}\\cdot {{({{y}^{4}})}^{2}}}[\/latex]<\/p>\r\nSeparate the squared factors into individual radicals.\r\n<p style=\"text-align: center;\">[latex] \\sqrt{{{7}^{2}}}\\cdot \\sqrt{{{({{x}^{5}})}^{2}}}\\cdot \\sqrt{{{({{y}^{4}})}^{2}}}[\/latex]<\/p>\r\nTake the square root of each radical using the rule that [latex] \\sqrt{{{x}^{2}}}=x[\/latex].\r\n<p style=\"text-align: center;\">[latex] 7\\cdot {{x}^{5}}\\cdot {{y}^{4}}[\/latex]<\/p>\r\nMultiply.\r\n<p style=\"text-align: center;\">[latex] 7{{x}^{5}}{{y}^{4}}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex] \\sqrt{49{{x}^{10}}{{y}^{8}}}=7|{{x}^{5}}|{{y}^{4}}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Simplify cube roots<\/h2>\r\nWe can use the same techniques we have used for simplifying square roots to simplify higher order roots. For example to simplify a cube root, the goal is to find factors under the radical that are perfect cubes\u00a0so that you can take their cube\u00a0root. We no longer need to be concerned about whether we have identified the principal root since we are now finding cube roots. Focus on finding identical trios\u00a0of factors as you simplify.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\sqrt[3]{40{{m}^{5}}}[\/latex]\r\n\r\n[reveal-answer q=\"617053\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"617053\"]Factor 40 into prime factors.\r\n\r\n[latex] \\sqrt[3]{5\\cdot 2\\cdot 2\\cdot 2\\cdot {{m}^{5}}}[\/latex]\r\n\r\nSince you are looking for the cube root, you need to find factors that appear 3 times under the radical. Rewrite [latex] 2\\cdot 2\\cdot 2[\/latex] as [latex] {{2}^{3}}[\/latex].\r\n\r\n[latex] \\sqrt[3]{{{2}^{3}}\\cdot 5\\cdot {{m}^{5}}}[\/latex]\r\n\r\nRewrite [latex] {{m}^{5}}[\/latex] as [latex] {{m}^{3}}\\cdot {{m}^{2}}[\/latex].\r\n\r\n[latex] \\sqrt[3]{{{2}^{3}}\\cdot 5\\cdot {{m}^{3}}\\cdot {{m}^{2}}}[\/latex]\r\n\r\nRewrite the expression as a product of multiple radicals.\r\n\r\n[latex] \\sqrt[3]{{{2}^{3}}}\\cdot \\sqrt[3]{5}\\cdot \\sqrt[3]{{{m}^{3}}}\\cdot \\sqrt[3]{{{m}^{2}}}[\/latex]\r\n\r\nSimplify and multiply.\r\n\r\n[latex] 2\\cdot \\sqrt[3]{5}\\cdot m\\cdot \\sqrt[3]{{{m}^{2}}}[\/latex]\r\n<h4>Answer<\/h4>\r\n[latex] \\sqrt[3]{40{{m}^{5}}}=2m\\sqrt[3]{5{{m}^{2}}}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nRemember that you can take the cube root of a negative expression. In the next example we will simplify a cube root with a negative radicand.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\sqrt[3]{-27{{x}^{4}}{{y}^{3}}}[\/latex]\r\n\r\n[reveal-answer q=\"670300\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"670300\"]Factor the expression into cubes.\r\n\r\nSeparate the cubed factors into individual radicals.\r\n\r\n[latex]\\begin{array}{r}\\sqrt[3]{-1\\cdot 27\\cdot {{x}^{4}}\\cdot {{y}^{3}}}\\\\\\sqrt[3]{{{(-1)}^{3}}\\cdot {{(3)}^{3}}\\cdot {{x}^{3}}\\cdot x\\cdot {{y}^{3}}}\\\\\\sqrt[3]{{{(-1)}^{3}}}\\cdot \\sqrt[3]{{{(3)}^{3}}}\\cdot \\sqrt[3]{{{x}^{3}}}\\cdot \\sqrt[3]{x}\\cdot \\sqrt[3]{{{y}^{3}}}\\end{array}[\/latex]\r\n\r\nSimplify the cube roots.\r\n\r\n[latex] -1\\cdot 3\\cdot x\\cdot y\\cdot \\sqrt[3]{x}[\/latex]\r\n<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\nYou could check your answer by performing the inverse operation. If you are right, when you cube [latex] -3xy\\sqrt[3]{x}[\/latex] you should get [latex] -27{{x}^{4}}{{y}^{3}}[\/latex].\r\n\r\n[latex] \\begin{array}{l}\\left( -3xy\\sqrt[3]{x} \\right)\\left( -3xy\\sqrt[3]{x} \\right)\\left( -3xy\\sqrt[3]{x} \\right)\\\\-3\\cdot -3\\cdot -3\\cdot x\\cdot x\\cdot x\\cdot y\\cdot y\\cdot y\\cdot \\sqrt[3]{x}\\cdot \\sqrt[3]{x}\\cdot \\sqrt[3]{x}\\\\-27\\cdot {{x}^{3}}\\cdot {{y}^{3}}\\cdot \\sqrt[3]{{{x}^{3}}}\\\\-27{{x}^{3}}{{y}^{3}}\\cdot x\\\\-27{{x}^{4}}{{y}^{3}}\\end{array}[\/latex]\r\n\r\nYou can also skip the step of factoring out the negative one once you are comfortable with identifying cubes.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\sqrt[3]{-24{{a}^{5}}}[\/latex]\r\n\r\n[reveal-answer q=\"473861\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"473861\"]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\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 following video we show more examples of simlifying cube roots.\r\n\r\nhttps:\/\/youtu.be\/BtJruOpmHCE\r\n<h2>Simplifying fourth roots<\/h2>\r\nNow let's move to simplifying fourth degree roots. \u00a0No matter what root you are simplifying, the same idea applies, find cubes for cube roots, powers of four for fourth roots, etc. Recall that when your simplified expression contains an even indexed radical and a variable factor with an odd exponent, you need to apply an absolute value.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\sqrt[4]{81{{x}^{8}}{{y}^{3}}}[\/latex]\r\n\r\n[reveal-answer q=\"295348\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"295348\"]Rewrite the expression.\r\n\r\n[latex] \\sqrt[4]{81}\\cdot \\sqrt[4]{{{x}^{8}}}\\cdot \\sqrt[4]{{{y}^{3}}}[\/latex]\r\n\r\nFactor each radicand.\r\n\r\n[latex] \\sqrt[4]{3\\cdot 3\\cdot 3\\cdot 3}\\cdot \\sqrt[4]{{{x}^{2}}\\cdot {{x}^{2}}\\cdot {{x}^{2}}\\cdot {{x}^{2}}}\\cdot \\sqrt[4]{{{y}^{3}}}[\/latex]\r\n\r\nSimplify.\r\n\r\n[latex]\\begin{array}{r}\\sqrt[4]{{{3}^{4}}}\\cdot \\sqrt[4]{{{({{x}^{2}})}^{4}}}\\cdot \\sqrt[4]{{{y}^{3}}}\\\\3\\cdot {{x}^{2}}\\cdot \\sqrt[4]{{{y}^{3}}}\\end{array}[\/latex]\r\n<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\nAn alternative method to factoring is to rewrite the expression with rational exponents, then use the rules of exponents to simplify. \u00a0You may find that you prefer one method over the other. Either way, it is nice to have options. We will show the last example again, using this idea.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\sqrt[4]{81{{x}^{8}}{{y}^{3}}}[\/latex]\r\n\r\n[reveal-answer q=\"324337\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"324337\"]Rewrite the radical using rational exponents.\r\n\r\n[latex] {{(81{{x}^{8}}{{y}^{3}})}^{\\frac{1}{4}}}[\/latex]\r\n\r\nUse the rules of exponents to simplify the expression.\r\n\r\n[latex] \\begin{array}{r}{{81}^{\\frac{1}{4}}}\\cdot {{x}^{\\frac{8}{4}}}\\cdot {{y}^{\\frac{3}{4}}}\\\\{{(3\\cdot 3\\cdot 3\\cdot 3)}^{\\frac{1}{4}}}{{x}^{2}}{{y}^{\\frac{3}{4}}}\\\\{{({{3}^{4}})}^{\\frac{1}{4}}}{{x}^{2}}{{y}^{\\frac{3}{4}}}\\\\3{{x}^{2}}{{y}^{\\frac{3}{4}}}\\end{array}[\/latex]\r\n\r\nChange the expression with the rational exponent back to radical form.\r\n\r\n[latex] 3{{x}^{2}}\\sqrt[4]{{{y}^{3}}}[\/latex]\r\n<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\nIn the following video we show another example of how to simplify a fourth and fifth root.\r\n\r\nhttps:\/\/youtu.be\/op2LEb0YRyw\r\n\r\nFor our last example, we will simplify\u00a0a more complicated expression, [latex]\\large\\frac{10{{b}^{2}}{{c}^{2}}}{c\\sqrt[3]{8{{b}^{4}}}}[\/latex]<i>.<\/i><i> <\/i>This expression has two variables, a fraction, and a radical. Let\u2019s take it step-by-step and see if using fractional exponents can help us simplify it.\r\nWe will\u00a0start by simplifying the denominator, since this is where the radical sign is located. Recall that an exponent in the denominator or a fraction can be rewritten as a negative exponent.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex]\\large\\frac{10{{b}^{2}}{{c}^{2}}}{c\\sqrt[3]{8{{b}^{4}}}}[\/latex]\r\n\r\n[reveal-answer q=\"962386\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"962386\"]Separate the factors in the denominator.\r\n\r\n[latex] \\frac{10{{b}^{2}}{{c}^{2}}}{c\\cdot \\sqrt[3]{8}\\cdot \\sqrt[3]{{{b}^{4}}}}[\/latex]\r\n\r\nTake the cube root of 8, which is 2.\r\n\r\n[latex] \\frac{10{{b}^{2}}{{c}^{2}}}{c\\cdot 2\\cdot \\sqrt[3]{{{b}^{4}}}}[\/latex]\r\n\r\nRewrite the radical using a fractional exponent.\r\n\r\n[latex] \\frac{10{{b}^{2}}{{c}^{2}}}{c\\cdot 2\\cdot {{b}^{\\frac{4}{3}}}}[\/latex]\r\n\r\nRewrite the fraction as a series of factors in order to cancel factors (see next step).\r\n\r\n[latex] \\frac{10}{2}\\cdot \\frac{{{c}^{2}}}{c}\\cdot \\frac{{{b}^{2}}}{{{b}^{\\frac{4}{3}}}}[\/latex]\r\n\r\nSimplify the constant and <i>c<\/i> factors.\r\n\r\n[latex] 5\\cdot c\\cdot \\frac{{{b}^{2}}}{{{b}^{\\frac{4}{3}}}}[\/latex]\r\n\r\nUse the rule of negative exponents,\u00a0<i>n<\/i><sup>-<\/sup><i><sup>x<\/sup><\/i><i>=<\/i>[latex] \\frac{1}{{{n}^{x}}}[\/latex], to rewrite [latex] \\frac{1}{{{b}^{\\tfrac{4}{3}}}}[\/latex] as [latex] {{b}^{-\\tfrac{4}{3}}}[\/latex].\r\n\r\n[latex] 5c{{b}^{2}}{{b}^{-\\ \\frac{4}{3}}}[\/latex]\r\n\r\nCombine the <i>b<\/i> factors by adding the exponents.\r\n\r\n[latex] 5c{{b}^{\\frac{2}{3}}}[\/latex]\r\n\r\nChange the expression with the fractional exponent back to radical form. By convention, an expression is not usually considered simplified if it has a fractional exponent or a radical in the denominator.\r\n\r\n[latex] 5c\\sqrt[3]{{{b}^{2}}}[\/latex]\r\n<h4>Answer<\/h4>\r\n[latex] \\frac{10{{b}^{2}}{{c}^{2}}}{c\\sqrt[3]{8{{b}^{4}}}}=5c\\sqrt[3]{{{b}^{2}}}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWell, that took a while, but you did it. You applied what you know about fractional exponents, negative exponents, and the rules of exponents to simplify the expression.\r\n\r\nIn our last video we show how to use rational exponents to simplify radical expressions.\r\n\r\nhttps:\/\/youtu.be\/CfxhFRHUq_M\r\n<h2>Summary<\/h2>\r\nA radical expression is a mathematical way of representing the <i>n<\/i>th root of a number. Square roots and cube roots are the most common radicals, but a root can be any number. To simplify radical expressions, look for exponential factors within the radical, and then use the property [latex] \\sqrt[n]{{{x}^{n}}}=x[\/latex] if <i>n<\/i> is odd, and [latex] \\sqrt[n]{{{x}^{n}}}=\\left| x \\right|[\/latex] if <i>n<\/i> is even to pull out quantities. All rules of integer operations and exponents apply when simplifying radical expressions.\r\n\r\nThe steps to consider when simplifying a radical are outlined below.\r\n<div class=\"textbox shaded\">\r\n<h3>Simplifying a radical<\/h3>\r\nWhen working with exponents and radicals:\r\n<ul>\r\n \t<li>If <i>n<\/i> is odd, [latex] \\sqrt[n]{{{x}^{n}}}=x[\/latex].<\/li>\r\n \t<li>If <i>n<\/i> is even, [latex] \\sqrt[n]{{{x}^{n}}}=\\left| x \\right|[\/latex]. (The absolute value accounts for the fact that if <i>x<\/i> is negative and raised to an even power, that number will be positive, as will the <i>n<\/i>th principal root of that number.)<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Summary<\/h2>\r\nThe square root of a number is the number which, when multiplied by itself, gives the original number. Principal square roots are always positive and the square root of 0 is 0. You can only take the square root of values that are greater than or equal to 0. The square root of a perfect square will be an integer. Other roots can be simplified by identifying factors that are perfect squares, cubes, etc. Nth roots can be approximated using trial and error or a calculator.\r\n\r\nAny radical in the form [latex]\\sqrt[n]{a^{x}}[\/latex]\u00a0 can be written using a fractional exponent in the form [latex]a^{\\frac{x}{n}}[\/latex]. Rewriting radicals using fractional exponents can be useful in simplifying some radical expressions. When working with fractional exponents, remember that fractional exponents are subject to all of the same rules as other exponents when they appear in algebraic expressions.\r\n<h2><\/h2>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Introduction to Roots\n<ul>\n<li>Define and evaluate principal square roots<\/li>\n<li>Define and evaluate nth roots<\/li>\n<li>Estimate roots that are not perfect<\/li>\n<\/ul>\n<\/li>\n<li>Radical Expressions and Rational Exponents\n<ul>\n<li>Define and identify a radical expression<\/li>\n<li>Convert radicals to expressions with rational exponents<\/li>\n<li>Convert expressions with rational exponents to their radical equivalent<\/li>\n<\/ul>\n<\/li>\n<li>Simplify Radical Expressions\n<ul>\n<li>Simplify radical expressions using factoring<\/li>\n<li>Simplify radical expressions\u00a0using rational exponents\u00a0and the laws of exponents<\/li>\n<li>Define [latex]\\sqrt{x^2}=|x|[\/latex], and apply it when simplifying radical expressions<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<p>Did you know that you can take the 6th root of a number? You have probably heard of a square root, written [latex]\\sqrt{}[\/latex], but you can also take a third, fourth and even a 5,000th root (if you really had to). In this lesson we will learn how a square root is defined and then we will build on that to form an understanding of nth roots. \u00a0We will use factoring and rules for exponents to simplify mathematical expressions that contain roots.<\/p>\n<p>The most common root is the <strong>square root<\/strong>. First, we will define what square roots are,\u00a0 and how you find the square root of a number. Then we will apply similar ideas to define and evaluate nth roots.<\/p>\n<p>Roots are the inverse of exponents, much like multiplication is the inverse of division. Recall\u00a0how exponents are defined, and written; with an exponent, as words, and as repeated multiplication.<\/p>\n<p><strong>Exponent:<\/strong> [latex]{{3}^{2}}[\/latex],\u00a0[latex]{{4}^{5}}[\/latex],\u00a0[latex]{{x}^{3}}[\/latex],\u00a0[latex]{{x}^{\\text{n}}}[\/latex]<\/p>\n<p><strong>Name:<\/strong>\u00a0\u201cThree squared\u201d or\u00a0\u201cThree to the second power\u201d,\u00a0\u201cFour to the fifth power\u201d,\u00a0\u201c<i>x<\/i> cubed\u201d,\u00a0\u201c<i>x<\/i> to the <i>n<\/i>th power\u201d<\/p>\n<p><strong>Repeated Multiplication:<\/strong>\u00a0[latex]3\\cdot 3[\/latex], \u00a0[latex]4\\cdot 4\\cdot 4\\cdot 4\\cdot 4[\/latex], \u00a0[latex]x\\cdot x\\cdot x[\/latex], \u00a0[latex]\\underbrace{x\\cdot x\\cdot x...\\cdot x}_{n\\text{ times}}[\/latex].<\/p>\n<p>Conversely,\u00a0 when you are trying to find the square root of a number (say, 25), you are trying to find a number that can be multiplied by itself to create that original number. In the case of 25, you can find that [latex]5\\cdot5=25[\/latex], so 5 must be the square root.<\/p>\n<h2>Square Roots<\/h2>\n<p>The symbol for the square root is called a <strong>radical symbol<\/strong> and looks like this: [latex]\\sqrt{\\,\\,\\,}[\/latex]. The expression [latex]\\sqrt{25}[\/latex] is read \u201cthe square root of twenty-five\u201d or \u201cradical twenty-five.\u201d The number that is written under the radical symbol is called the <strong>radicand<\/strong>.<br \/>\n<img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200220\/CNX_CAT_Figure_01_03_002.jpg\" alt=\"The expression: square root of twenty-five is enclosed in a circle. The circle has an arrow pointing to it labeled: Radical expression. The square root symbol has an arrow pointing to it labeled: Radical. The number twenty-five has an arrow pointing to it labeled: Radicand.\" data-media-type=\"image\/jpg\" \/><\/p>\n<p>The following table shows different radicals and their equivalent written and simplified forms.<\/p>\n<table style=\"width: 70%;\">\n<thead>\n<tr>\n<th>Radical<\/th>\n<th>Name<\/th>\n<th>Simplified Form<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]\\sqrt{36}[\/latex]<\/td>\n<td>\n<p>\u201cSquare root of thirty-six\u201d<\/p>\n<p>\u201cRadical thirty-six\u201d<\/td>\n<td>[latex]\\sqrt{36}=\\sqrt{6\\cdot 6}=6[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\sqrt{100}[\/latex]<\/td>\n<td>\n<p>\u201cSquare root of one hundred\u201d<\/p>\n<p>\u201cRadical one hundred\u201d<\/td>\n<td>[latex]\\sqrt{100}=\\sqrt{10\\cdot 10}=10[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\sqrt{225}[\/latex]<\/td>\n<td>\n<p>\u201cSquare root of two hundred twenty-five\u201d<\/p>\n<p>\u201cRadical two hundred twenty-five\u201d<\/td>\n<td>[latex]\\sqrt{225}=\\sqrt{15\\cdot 15}=15[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Consider [latex]\\sqrt{25}[\/latex] again. You may realize that there is another value that, when multiplied by itself, also results in 25. That number is [latex]\u22125[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}5\\cdot 5=25\\\\-5\\cdot -5=25\\end{array}[\/latex]<\/p>\n<p>By definition, the square root symbol always means to find the positive root, called the <strong>principal root<\/strong>. So while [latex]5\\cdot5[\/latex] and [latex]\u22125\\cdot\u22125[\/latex] both equal 25, only 5 is the principal root. You should also know that zero is special because it has only one square root: itself (since [latex]0\\cdot0=0[\/latex]).<\/p>\n<p>In our first example we will show you how to use radical notation to evaluate principal square roots.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Find the principal root of each expression.<\/p>\n<ol>\n<li>[latex]\\sqrt{100}[\/latex]<\/li>\n<li>[latex]\\sqrt{16}[\/latex]<\/li>\n<li>[latex]\\sqrt{25+144}[\/latex]<\/li>\n<li>[latex]\\sqrt{49}-\\sqrt{81}\\\\[\/latex]<\/li>\n<li>[latex]-\\sqrt{81}[\/latex]<\/li>\n<li>[latex]\\sqrt{-9}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q419579\">Show Answer<\/span><\/p>\n<div id=\"q419579\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]\\sqrt{100}=10[\/latex] because [latex]{10}^{2}=100[\/latex]<\/li>\n<li>[latex]\\sqrt{\\sqrt{16}}=\\sqrt{4}=2[\/latex] because [latex]{4}^{2}=16[\/latex] and [latex]{2}^{2}=4[\/latex]<\/li>\n<li>Recall that square roots act as grouping symbols in the order of operations, so addition and subtraction must be performed first when they occur under a radical. [latex]\\sqrt{25+144}=\\sqrt{169}=13[\/latex] because [latex]{13}^{2}=169[\/latex]<\/li>\n<li>This problem is similar to the last one, but this time subtraction should occur after evaluating the root. Stop and think about why these two problems are different. [latex]\\sqrt{49}-\\sqrt{81}=7 - 9=-2[\/latex] because [latex]{7}^{2}=49[\/latex] and [latex]{9}^{2}=81[\/latex]<\/li>\n<li>\n<p style=\"text-align: left;\">The negative in front means to take the opposite of the value after you simplify the radical. [latex]-\\sqrt{81}\\\\-\\sqrt{9\\cdot 9}[\/latex].\u00a0 The square root of 81 is 9. Then, take the opposite of 9. [latex]\u2212(9)[\/latex]<\/p>\n<\/li>\n<li>[latex]\\sqrt{-9}[\/latex], we are looking for a number that when it is squared, returns [latex]-9[\/latex]. We can try [latex](-3)^2[\/latex], but that will give a positive result, and [latex]3^2[\/latex] will also give a positive result. This leads to an important fact &#8211; \u00a0you cannot find the square root of a negative number.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video we present more examples of how to find a principle square root.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Simplify a Variety of Square Expressions (Simplify Perfectly)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/2cWAkmJoaDQ?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>The last example we showed leads to an important characteristic of square roots. You can only take the square root of values that are nonnegative.<\/p>\n<p class=\"textbox shaded\"><strong>Domain of a Square Root<\/strong><br \/>\n[latex]\\sqrt{-a}[\/latex] is not defined for all real numbers, a. Therefore, [latex]\\sqrt{a}[\/latex] is defined for [latex]a\\ge0[\/latex]<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Think About It<\/h3>\n<p>Does [latex]\\sqrt{25}=\\pm 5[\/latex]? Write your ideas and a sentence to defend them in the box below before you look at the answer.<\/p>\n<p><textarea aria-label=\"Your Answer\" rows=\"1\"><\/textarea><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q101071\">Show Answer<\/span><\/p>\n<div id=\"q101071\" class=\"hidden-answer\" style=\"display: none\">\n<p><em>No. Although both<\/em> [latex]{5}^{2}[\/latex] <em>and<\/em> [latex]{\\left(-5\\right)}^{2}[\/latex] <em>are<\/em> [latex]25[\/latex], <em>the radical symbol implies only a nonnegative root, the principal square root. The principal square root of 25 is<\/em> [latex]\\sqrt{25}=5[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Cube Roots<\/h2>\n<p>We know that [latex]5^2=25, \\text{ and }\\sqrt{25}=5[\/latex] but what if we want to &#8220;undo&#8221; [latex]5^3=125, \\text{ or }5^4=625[\/latex]? We can use higher order roots to answer these questions.<\/p>\n<p>Suppose we know that [latex]{a}^{3}=8[\/latex]. We want to find what number raised to the 3rd power is equal to 8. Since [latex]{2}^{3}=8[\/latex], we say that 2 is the cube root of 8. In the next example we will evaluate the cube roots of some perfect cubes.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Evaluate the following:<\/p>\n<ol>\n<li>[latex]\\sqrt[3]{125}[\/latex]<\/li>\n<li>[latex]\\sqrt[3]{-8}[\/latex]<\/li>\n<li>[latex]\\sqrt[3]{27}[\/latex]<\/li>\n<\/ol>\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\">\n<p>1. You can read this as \u201cthe third root of 125\u201d or \u201cthe cube root of 125.\u201d To evaluate this expression, look for a number that, when multiplied by itself two times (for a total of three identical factors), equals 125. [latex]\\text{?}\\cdot\\text{?}\\cdot\\text{?}=125[\/latex]. Since 125 ends in 5, 5 is a good candidate. [latex]5\/cdot5\/cdot5=125[\/latex]<br \/>\n2. We want to find a number whose cube is 8. [latex]2\\cdot2\\cdot2=8[\/latex] the cube root of 8 is 2.<\/p>\n<p>3. 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.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>As we saw in the last example,there is one interesting fact about cube roots that is not true of square roots. Negative numbers can\u2019t have real number square roots, but negative numbers can have real number cube roots! What is the cube root of [latex]\u22128[\/latex]? [latex]\\sqrt[3]{-8}=-2[\/latex] because [latex]-2\\cdot -2\\cdot -2=-8[\/latex]. Remember, when you are multiplying an odd number of negative numbers, the result is negative! Consider [latex]\\sqrt[3]{{{(-1)}^{3}}}=-1[\/latex].<\/p>\n<p>In the following video we show more examples of finding a cube root.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" 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<h2>Nth Roots<\/h2>\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<p>We can apply the same idea to any exponent and it&#8217;s corresponding root.\u00a0 The <em>n<\/em>th root of [latex]a[\/latex] is a number that, when raised to the <em>n<\/em>th power, gives [latex]a[\/latex]. For example, [latex]3[\/latex] is the 5th root of [latex]243[\/latex] because [latex]{\\left(3\\right)}^{5}=243[\/latex]. If [latex]a[\/latex] is a real number with at least one <em>n<\/em>th root, then the <strong>principal <em>n<\/em>th root<\/strong> of [latex]a[\/latex] is the number with the same sign as [latex]a[\/latex] that, when raised to the <em>n<\/em>th power, equals [latex]a[\/latex].<\/p>\n<p>The principal <em>n<\/em>th root of [latex]a[\/latex] is written as [latex]\\sqrt[n]{a}[\/latex], where [latex]n[\/latex] is a positive integer greater than or equal to 2. In the radical expression, [latex]n[\/latex] is called the <strong>index<\/strong> of the radical.<\/p>\n<div class=\"textbox\">\n<h3>Definition:\u00a0Principal <em>n<\/em>th Root<\/h3>\n<p>If [latex]a[\/latex] is a real number with at least one <em>n<\/em>th root, then the <strong>principal <em>n<\/em>th root<\/strong> of [latex]a[\/latex], written as [latex]\\sqrt[n]{a}[\/latex], is the number with the same sign as [latex]a[\/latex] that, when raised to the <em>n<\/em>th power, equals [latex]a[\/latex]. The <strong>index<\/strong> of the radical is [latex]n[\/latex].<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Evaluate each of the following:<\/p>\n<ol>\n<li>[latex]\\sqrt[5]{-32}[\/latex]<\/li>\n<li>[latex]\\sqrt[4]{81}[\/latex]<\/li>\n<li>[latex]\\sqrt[8]{-1}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q140298\">Show Answer<\/span><\/p>\n<div id=\"q140298\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]\\sqrt[5]{-32}[\/latex] Factor 32, because [latex]{\\left(-2\\right)}^{5}=-32 \\\\ \\text{ }[\/latex]<\/li>\n<li>[latex]\\sqrt[4]{81}[\/latex]. Factoring can help, we know that [latex]9\\cdot9=81[\/latex] and we can further factor each 9: [latex]\\sqrt[4]{81}=\\sqrt[4]{3\\cdot3\\cdot3\\cdot3}=\\sqrt[4]{3^4}=3[\/latex]<\/li>\n<li>[latex]\\sqrt[8]{-1}[\/latex], since we have an 8th root &#8211; which is even- with a negative number as the radicand, this root has no real number solutions. In other words, [latex]-1\\cdot-1\\cdot-1\\cdot-1\\cdot-1\\cdot-1\\cdot-1\\cdot-1=+1[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video we show more examples of how to evaluate and nth root.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Simplify Perfect Nth Roots\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/vA2DkcUSRSk?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/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 evaluate the radicals [latex]\\sqrt[3]{-81},\\ \\sqrt[5]{-64}[\/latex], and [latex]\\sqrt[7]{-2187}[\/latex], but you cannot evaluate the radicals [latex]\\sqrt[{}]{-100},\\ \\sqrt[4]{-16}[\/latex], or [latex]\\sqrt[6]{-2,500}[\/latex].<\/p>\n<h2>Estimate Roots<\/h2>\n<p>An approach to handling roots that are not perfect (squares, cubes, etc.)\u00a0 is to approximate them by comparing the values to perfect squares, cubes, or nth roots. Suppose you wanted to know the square root of 17. Let\u2019s look at how you might approximate it.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Estimate. [latex]\\sqrt{17}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q358591\">Show Solution<\/span><\/p>\n<div id=\"q358591\" class=\"hidden-answer\" style=\"display: none\">Think of two perfect squares that surround 17.\u00a017 is in between the perfect squares 16 and 25.\u00a0So, [latex]\\sqrt{17}[\/latex] must be in between [latex]\\sqrt{16}[\/latex] and [latex]\\sqrt{25}[\/latex].<\/p>\n<p>Determine whether [latex]\\sqrt{17}[\/latex] is closer to 4 or to 5 and make another estimate.<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt{16}=4[\/latex] and [latex]\\sqrt{25}=5[\/latex]<\/p>\n<p>Since 17 is closer to 16 than 25, [latex]\\sqrt{17}[\/latex] is probably about 4.1 or 4.2.<\/p>\n<p>Use trial and error to get a better estimate of [latex]\\sqrt{17}[\/latex]. Try squaring incrementally greater numbers, beginning with 4.1, to find a good approximation for [latex]\\sqrt{17}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\left(4.1\\right)^{2}[\/latex]<\/p>\n<p>[latex]\\left(4.1\\right)^{2}[\/latex]\u00a0gives a closer estimate than [latex](4.2)^{2}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]4.1\\cdot4.1=16.81\\\\4.2\\cdot4.2=17.64[\/latex]<\/p>\n<p>Continue to use trial and error to get an even better estimate.<\/p>\n<p style=\"text-align: center;\">[latex]4.12\\cdot4.12=16.9744\\\\4.13\\cdot4.13=17.0569[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\sqrt{17}\\approx 4.12[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>This approximation is pretty close. If you kept using this trial and error strategy you could continue to find the square root to the thousandths, ten-thousandths, and hundred-thousandths places, but eventually it would become too tedious to do by hand.<\/p>\n<p>For this reason, when you need to find a more precise approximation of a square root, you should use a calculator. Most calculators have a square root key [latex](\\sqrt{{}})[\/latex] that will give you the square root approximation quickly. On a simple 4-function calculator, you would likely key in the number that you want to take the square root of and then press the square root key.<\/p>\n<p>Try to find [latex]\\sqrt{17}[\/latex] using your calculator. Note that you will not be able to get an \u201cexact\u201d answer because [latex]\\sqrt{17}[\/latex] is an irrational number, a number that cannot be expressed as a fraction, and the decimal never terminates or repeats. To nine decimal positions, [latex]\\sqrt{17}[\/latex] is approximated as 4.123105626. A calculator can save a lot of time and yield a more precise square root when you are dealing with numbers that aren\u2019t perfect squares.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Approximate [latex]\\sqrt[3]{30}[\/latex] and also find its value using a calculator.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q71092\">Show Solution<\/span><\/p>\n<div id=\"q71092\" class=\"hidden-answer\" style=\"display: none\">Find the cubes that surround 30.<\/p>\n<p>30 is inbetween the perfect cubes 27 and 81.<\/p>\n<p>[latex]\\sqrt[3]{27}=3[\/latex] and [latex]\\sqrt[3]{81}=4[\/latex], so [latex]\\sqrt[3]{30}[\/latex] is between 3 and 4.<br \/>\nUse a calculator.<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt[3]{30}\\approx3.10723[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>By approximation: [latex]3\\ge\\sqrt[3]{30}\\le4[\/latex]<\/p>\n<p>Using a calculator: [latex]\\sqrt[3]{30}\\approx3.10723[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The following video shows another example of how to estimate a square root.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" title=\"Approximate a Square Root to Two Decimal Places Using Trial and Error\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/iNfalyW7olk?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Radical Expressions and Rational Exponents<\/h2>\n<p>Square roots are most often written using a radical sign, like this, [latex]\\sqrt{4}[\/latex]. But there is another way to represent them. You can use rational exponents instead of a radical. A <strong>rational exponent<\/strong> is an exponent that is a fraction. For example, [latex]\\sqrt{4}[\/latex] can be written as [latex]{{4}^{\\tfrac{1}{2}}}[\/latex].<\/p>\n<p>Can\u2019t imagine raising a number to a rational exponent? They may be hard to get used to, but rational exponents can actually help simplify some problems. Writing radicals with rational exponents will come in handy when we discuss techniques for simplifying more complex radical expressions.<\/p>\n<p><strong>Radical expressions<\/strong> are expressions that contain radicals. Radical expressions come in many forms, from simple and familiar, such as[latex]\\sqrt{16}[\/latex], to quite complicated, as in [latex]\\sqrt[3]{250{{x}^{4}}y}[\/latex]<\/p>\n<h2>Write an expression with a rational exponent as a radical<\/h2>\n<p>Radicals and fractional exponents are alternate ways of expressing the same thing. \u00a0In the table below we show equivalent ways to express radicals: with a root, with a rational exponent, and as a principal root.<\/p>\n<table style=\"width: 30%;\">\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;\">Principal Root<\/p>\n<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"text-align: center;\">[latex]\\sqrt{16}[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]{{16}^{\\tfrac{1}{2}}}[\/latex]<\/td>\n<td style=\"text-align: center;\">4<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]\\sqrt{25}[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]{{25}^{\\tfrac{1}{2}}}[\/latex]<\/td>\n<td style=\"text-align: center;\">5<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]\\sqrt{100}[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]{{100}^{\\tfrac{1}{2}}}[\/latex]<\/td>\n<td style=\"text-align: center;\">10<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Let\u2019s look at some more examples, but this time with cube roots. Remember, cubing a number raises it to the power of three. Notice that in the examples in the table below, the denominator of the rational exponent is the number 3.<\/p>\n<table style=\"width: 30%;\">\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;\">Principal Root<\/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: 30%;\">\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<p>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]{{}}[\/latex], and [latex]\\frac{1}{8}[\/latex] translates to the eighth root or [latex]\\sqrt[8]{{}}[\/latex].<\/p>\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}{3}}}}[\/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[3]{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}{3}}}}=2\\sqrt[3]{x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Write a radical expression as an expression with a rational exponent<\/h2>\n<div id=\"attachment_3123\" style=\"width: 151px\" class=\"wp-caption alignright\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3123\" class=\"wp-image-3123\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/26174543\/Screen-Shot-2016-07-26-at-10.44.01-AM-300x291.png\" alt=\"Person sitting on the ground with one leg arched behind them and one leg curved in front of them.\" width=\"141\" height=\"137\" \/><\/p>\n<p id=\"caption-attachment-3123\" class=\"wp-caption-text\">Flexibility<\/p>\n<\/div>\n<p>We can write radicals with rational exponents, and as we will see when we simplify more complex radical expressions, this can make things easier. Having different ways to express and write algebraic expressions allows us to have flexibility in solving and simplifying them. It is like having a thesaurus when you write, you want to have options for expressing yourself!<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Write [latex]\\sqrt[4]{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]\\Large\\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>[latex]{{81}^{\\frac{1}{4}}}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\sqrt[4]{81}={{81}^{\\frac{1}{4}}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\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<h2>Rational exponents whose numerator is not equal to one<\/h2>\n<p>All of the numerators for the fractional exponents in the examples above were 1. You can use fractional exponents that have numerators other than 1 to express roots, as shown below.<\/p>\n<table>\n<thead>\n<tr>\n<th>\n<p style=\"text-align: center;\">Radical<\/p>\n<\/th>\n<th>\n<p style=\"text-align: center;\">Exponent<\/p>\n<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"text-align: center;\">[latex]\\sqrt{9}[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]9^{\\frac{1}{2}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]\\sqrt[3]{{{9}^{2}}}[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]9^{\\frac{2}{3}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]\\sqrt[4]{9^{3}}[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]9^{\\frac{3}{4}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]\\sqrt[5]{9^{2}}[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]9^{\\frac{2}{5}}[\/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]{9^{x}}[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]9\\frac{x}{n}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3198 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/29225734\/Screen-Shot-2016-07-29-at-3.56.45-PM-300x179.png\" alt=\"Screen Shot 2016-07-29 at 3.56.45 PM\" width=\"380\" height=\"227\" \/><\/p>\n<p>To rewrite a radical using a fractional exponent, the power to which the radicand is raised becomes the numerator and the root\/ index becomes the denominator.<\/p>\n<div class=\"textbox shaded\">\n<h3>Writing Rational\u00a0Exponents<\/h3>\n<p>Any radical in the form [latex]\\sqrt[n]{a^{x}}[\/latex]\u00a0 can be written using a fractional exponent in the form [latex]a^{\\frac{x}{n}}[\/latex].<\/p>\n<\/div>\n<p>The relationship between [latex]\\sqrt[n]{{{a}^{x}}}[\/latex]and [latex]{{a}^{\\frac{x}{n}}}[\/latex] works for rational exponents that have a numerator of 1 as well. For example, the radical [latex]\\sqrt[3]{8}[\/latex] can also be written as [latex]\\sqrt[3]{{{8}^{1}}}[\/latex], since any number remains the same value if it is raised to the first power. You can now see where the numerator of 1 comes from in the equivalent form of [latex]{{8}^{\\frac{1}{3}}}[\/latex].<\/p>\n<p>In the next example, we practice writing radicals with rational exponents where the numerator is not equal to one.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Rewrite the radicals using a rational exponent, then simplify your result.<\/p>\n<ol>\n<li>[latex]\\sqrt[3]{{{a}^{6}}}[\/latex]<\/li>\n<li>[latex]\\sqrt[12]{16^3}[\/latex]<\/li>\n<\/ol>\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\">\n<p>1.[latex]\\sqrt[n]{a^{x}}[\/latex] can be rewritten as\u00a0[latex]a^{\\frac{x}{n}}[\/latex], so in this case [latex]n=3,\\text{ and }x=6[\/latex], therefore<\/p>\n<p>[latex]\\sqrt[3]{{{a}^{6}}}={{a}^{\\frac{6}{3}}}[\/latex]<\/p>\n<p>Simplify the exponent.<\/p>\n<p>[latex]{{a}^{\\frac{6}{3}}}={{a}^{2}}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\sqrt[3]{{{a}^{6}}}={{a}^{2}}[\/latex]<\/p>\n<p>2.\u00a0[latex]\\sqrt[n]{a^{x}}[\/latex] can be rewritten as\u00a0[latex]a^{\\frac{x}{n}}[\/latex], so in this case [latex]n=12,\\text{ and }x=3[\/latex], therefore<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt[12]{16^3}={16}^{\\frac{3}{12}}={16}^{\\frac{1}{4}}[\/latex]<\/p>\n<p>Simplify the expression using rules for exponents.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ccc}16=2^4\\\\{16}^{\\frac{1}{4}}={2^4}^{\\frac{1}{4}}\\\\=2^{4\\cdot\\frac{1}{4}}\\\\=2^1=2\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\sqrt[12]{16^3}=2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In our last example we will rewrite expressions with rational exponents as radicals. This practice will help us when we simplify more complicated radical expressions, and as we learn how to solve radical equations. Typically it is easier to simplify when we use rational exponents, but this exercise is intended to help you understand how\u00a0the numerator and denominator of the exponent are\u00a0the exponent of a radicand and index of a radical.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Rewrite the expressions\u00a0using a radical.<\/p>\n<ol>\n<li>[latex]{x}^{\\frac{2}{3}}[\/latex]<\/li>\n<li>[latex]{5}^{\\frac{4}{7}}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q200228\">Show Answer<\/span><\/p>\n<div id=\"q200228\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]{x}^{\\frac{2}{3}}[\/latex], the numerator is 2 and the denominator is 3, therefore we will have the third root of x squared, [latex]\\sqrt[3]{x^2}[\/latex]<\/li>\n<li>[latex]{5}^{\\frac{4}{7}}[\/latex], the numerator is 4 and the denominator is 7, so we will have the seventh root of 5 raised to the fourth power. [latex]\\sqrt[7]{5^4}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video we show more examples of writing radical expressions with rational exponents and expressions with rational exponents as radical expressions.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-5\" 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>We will use this notation later, so come back for practice if you forget how\u00a0to write a radical with a rational exponent.<\/p>\n<h2>Simplify Radical Expressions<\/h2>\n<p><strong>Radical expressions<\/strong> are expressions that contain radicals. Radical expressions come in many forms, from simple and familiar, such as[latex]\\sqrt{16}[\/latex], to quite complicated, as in [latex]\\sqrt[3]{250{{x}^{4}}y}[\/latex].<\/p>\n<p>To simplify complicated radical expressions, we can use some definitions and rules from simplifying exponents. Recall the <strong>Product Raised to a Power Rule<\/strong> from when you studied exponents. This rule states that the product of two or more non-zero numbers raised to a power is equal to the product of each number raised to the same power. In math terms, it is written [latex]\\left(ab\\right)^{x}=a^{x}\\cdot{b}^{x}[\/latex].\u00a0So, for example, you can use the rule to rewrite [latex]{{\\left( 3x \\right)}^{2}}[\/latex] as [latex]{{3}^{2}}\\cdot {{x}^{2}}=9\\cdot {{x}^{2}}=9{{x}^{2}}[\/latex].<\/p>\n<p>Now instead of using the exponent 2, let\u2019s use the exponent [latex]\\frac{1}{2}[\/latex]. The exponent is distributed in the same way.<\/p>\n<p style=\"text-align: center;\">[latex]{{\\left( 3x \\right)}^{\\frac{1}{2}}}={{3}^{\\frac{1}{2}}}\\cdot {{x}^{\\frac{1}{2}}}[\/latex]<\/p>\n<p>And since you know that raising a number to the [latex]\\frac{1}{2}[\/latex] power is the same as taking the square root of that number, you can also write it this way.<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt{3x}=\\sqrt{3}\\cdot \\sqrt{x}[\/latex]<\/p>\n<p>Look at that\u2014you can think of any number underneath a radical as the <i>product of separate factors<\/i>, each underneath its own radical.<\/p>\n<div class=\"textbox shaded\">\n<h3>A Product Raised to a Power Rule\u00a0or sometimes called\u00a0The Square Root of a Product Rule<\/h3>\n<p>For any real numbers <i>a<\/i> and <i>b<\/i>, [latex]\\sqrt{ab}=\\sqrt{a}\\cdot \\sqrt{b}[\/latex].<\/p>\n<p>For example: [latex]\\sqrt{100}=\\sqrt{10}\\cdot \\sqrt{10}[\/latex], and [latex]\\sqrt{75}=\\sqrt{25}\\cdot \\sqrt{3}[\/latex]<\/p>\n<\/div>\n<p>This rule is important because it helps you think of one radical as the product of multiple radicals. If you can identify perfect squares within a radical, as with [latex]\\sqrt{(2\\cdot 2)(2\\cdot 2)(3\\cdot 3})[\/latex], you can rewrite the expression as the product of multiple perfect squares: [latex]\\sqrt{{{2}^{2}}}\\cdot \\sqrt{{{2}^{2}}}\\cdot \\sqrt{{{3}^{2}}}[\/latex].<\/p>\n<p class=\"p1\">The square root of a product rule will help us simplify roots that aren&#8217;t perfect, as is shown the following example.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\sqrt{63}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q908978\">Show Solution<\/span><\/p>\n<div id=\"q908978\" class=\"hidden-answer\" style=\"display: none\">63 is not a perfect square so we can use the\u00a0square root of a product rule to simplify any factors that are perfect squares.<br \/>\nFactor 63 into 7 and 9.<br \/>\n[latex]\\sqrt{7\\cdot 9}[\/latex]<br \/>\n9 is a perfect square, [latex]9=3^2[\/latex], therefore we can rewrite the radicand.<\/p>\n<p>[latex]\\sqrt{7\\cdot {{3}^{2}}}[\/latex]<\/p>\n<p>Using the Product Raised to a Power rule, separate the radical into the product of two factors, each under a radical.<br \/>\n[latex]\\sqrt{7}\\cdot \\sqrt{{{3}^{2}}}[\/latex]<br \/>\nTake the square root of [latex]3^{2}[\/latex].<br \/>\n[latex]\\sqrt{7}\\cdot 3[\/latex]<br \/>\nRearrange factors so the integer appears before the radical, and then multiply. (This is done so that it is clear that only the 7 is under the radical, not the 3.)<br \/>\n[latex]3\\cdot \\sqrt{7}[\/latex]<br \/>\n<b>Answer<\/b><br \/>\n[latex]\\sqrt{63}=3\\sqrt{7}[\/latex]\n<\/div>\n<\/div>\n<\/div>\n<p>The final answer [latex]3\\sqrt{7}[\/latex] may look a bit odd, but it is in simplified form. You can read this as \u201cthree radical seven\u201d or \u201cthree times the square root of seven.\u201d<\/p>\n<p>The following video shows more examples of how to simplify square roots that do not have perfect square radicands.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-6\" title=\"Simplify Square Roots (Not Perfect Square Radicands)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/oRd7aBCsmfU?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>Before we move on to simplifying more complex radicals with variables, we need to learn about an important behavior of square roots with variables in the radicand.<\/p>\n<p>Consider the expression [latex]\\sqrt{{{x}^{2}}}[\/latex]. This looks like it should be equal to <i>x<\/i>, right? 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 style=\"width: 40%;\">\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]! However, in all cases [latex]\\sqrt{x^{2}}=\\left|x\\right|[\/latex].\u00a0You need to consider this fact when simplifying radicals with an even index that contain variables, because by definition [latex]\\sqrt{x^{2}}[\/latex]\u00a0is always nonnegative.<\/p>\n<div class=\"textbox shaded\">\n<h3>Taking the Square Root of a Radical Expression<\/h3>\n<p>When finding the square root of an expression that contains variables raised to 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>We will combine this with the square root of a product rule in our next example to simplify an expression with three variables in the radicand.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\sqrt{{{a}^{3}}{{b}^{5}}{{c}^{2}}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q141094\">Show Solution<\/span><\/p>\n<div id=\"q141094\" class=\"hidden-answer\" style=\"display: none\">Factor to find variables with even exponents.<\/p>\n<p>[latex]\\sqrt{{{a}^{2}}\\cdot a\\cdot {{b}^{4}}\\cdot b\\cdot {{c}^{2}}}[\/latex]<\/p>\n<p>Rewrite [latex]b^{4}[\/latex]\u00a0as [latex]\\left(b^{2}\\right)^{2}[\/latex].<\/p>\n<p>[latex]\\sqrt{{{a}^{2}}\\cdot a\\cdot {{({{b}^{2}})}^{2}}\\cdot b\\cdot {{c}^{2}}}[\/latex]<\/p>\n<p>Separate the squared factors into individual radicals.<\/p>\n<p>[latex]\\sqrt{{{a}^{2}}}\\cdot \\sqrt{{{({{b}^{2}})}^{2}}}\\cdot \\sqrt{{{c}^{2}}}\\cdot \\sqrt{a\\cdot b}[\/latex]<\/p>\n<p>Take the square root of each radical. Remember that [latex]\\sqrt{{{a}^{2}}}=\\left| a \\right|[\/latex].<\/p>\n<p>[latex]\\left| a \\right|\\cdot {{b}^{2}}\\cdot \\left|{c}\\right|\\cdot \\sqrt{a\\cdot b}[\/latex]<\/p>\n<p>Simplify and multiply.<\/p>\n<p>[latex]\\left| ac \\right|{{b}^{2}}\\sqrt{ab}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\sqrt{{{a}^{3}}{{b}^{5}}{{c}^{2}}}=\\left| ac \\right|{{b}^{2}}\\sqrt{ab}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h3>Analysis of the Solution<\/h3>\n<p>Why didn&#8217;t we write [latex]b^2[\/latex] as [latex]|b^2|[\/latex]? \u00a0Because when you square a number, you will always get a positive result, so the principal square root of\u00a0[latex]\\left(b^2\\right)^2[\/latex] will always be non-negative. One tip for\u00a0knowing when to apply the absolute value after simplifying any even indexed root is to look at the final exponent on your variable terms. \u00a0If the exponent is odd &#8211; including 1 &#8211; add an absolute value. This applies to simplifying any root with an even index, as we will see in later examples.<\/p>\n<p>In the following video you will see more examples of how to simplify radical expressions with variables.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-7\" title=\"Simplify Square Roots with Variables\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/q7LqsKPoAKo?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>We will show another example where the simplified expression contains variables with both odd and even powers.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\sqrt{9{{x}^{6}}{{y}^{4}}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q41297\">Show Solution<\/span><\/p>\n<div id=\"q41297\" class=\"hidden-answer\" style=\"display: none\">Factor to find identical pairs.<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt{3\\cdot 3\\cdot {{x}^{3}}\\cdot {{x}^{3}}\\cdot {{y}^{2}}\\cdot {{y}^{2}}}[\/latex]<\/p>\n<p>Rewrite the pairs as perfect squares.<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt{{{3}^{2}}\\cdot {{\\left( {{x}^{3}} \\right)}^{2}}\\cdot {{\\left( {{y}^{2}} \\right)}^{2}}}[\/latex]<\/p>\n<p>Separate into individual radicals.<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt{{{3}^{2}}}\\cdot \\sqrt{{{\\left( {{x}^{3}} \\right)}^{2}}}\\cdot \\sqrt{{{\\left( {{y}^{2}} \\right)}^{2}}}[\/latex]<\/p>\n<p>Simplify.<\/p>\n<p style=\"text-align: center;\">[latex]3{{x}^{3}}{{y}^{2}}[\/latex]<\/p>\n<p style=\"text-align: left;\">Because x has an odd power, we will add the absolute value for our final solution.<\/p>\n<p style=\"text-align: center;\">[latex]3|{{x}^{3}}|{{y}^{2}}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\sqrt{9{{x}^{6}}{{y}^{4}}}=3|{{x}^{3}}|{y}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In our next example we will start with an expression written with a rational exponent. You will see that you can use a similar process &#8211; factoring and sorting terms into squares &#8211; to simplify this expression.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]{{(36{{x}^{4}})}^{\\frac{1}{2}}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q554375\">Show Solution<\/span><\/p>\n<div id=\"q554375\" class=\"hidden-answer\" style=\"display: none\">Rewrite the expression with the fractional exponent as a radical.<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt{36{{x}^{4}}}[\/latex]<\/p>\n<p>Find the square root of both the coefficient and the variable.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r} \\sqrt{{{6}^{2}}\\cdot {{x}^{4}}}\\\\\\sqrt{{{6}^{2}}}\\cdot \\sqrt{{{x}^{4}}}\\\\\\sqrt{{{6}^{2}}}\\cdot \\sqrt{{{({{x}^{2}})}^{2}}}\\\\6\\cdot{x}^{2}\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]{{(36{{x}^{4}})}^{\\frac{1}{2}}}=6{{x}^{2}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Here is one more example with perfect squares.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\sqrt{49{{x}^{10}}{{y}^{8}}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q283065\">Show Solution<\/span><\/p>\n<div id=\"q283065\" class=\"hidden-answer\" style=\"display: none\">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 style=\"text-align: center;\">[latex]\\sqrt{7\\cdot 7\\cdot {{x}^{5}}\\cdot {{x}^{5}}\\cdot {{y}^{4}}\\cdot {{y}^{4}}}[\/latex]<\/p>\n<p>Rewrite the pairs as squares.<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt{{{7}^{2}}\\cdot {{({{x}^{5}})}^{2}}\\cdot {{({{y}^{4}})}^{2}}}[\/latex]<\/p>\n<p>Separate the squared factors into individual radicals.<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt{{{7}^{2}}}\\cdot \\sqrt{{{({{x}^{5}})}^{2}}}\\cdot \\sqrt{{{({{y}^{4}})}^{2}}}[\/latex]<\/p>\n<p>Take the square root of each radical using the rule that [latex]\\sqrt{{{x}^{2}}}=x[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]7\\cdot {{x}^{5}}\\cdot {{y}^{4}}[\/latex]<\/p>\n<p>Multiply.<\/p>\n<p style=\"text-align: center;\">[latex]7{{x}^{5}}{{y}^{4}}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\sqrt{49{{x}^{10}}{{y}^{8}}}=7|{{x}^{5}}|{{y}^{4}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Simplify cube roots<\/h2>\n<p>We can use the same techniques we have used for simplifying square roots to simplify higher order roots. For example to simplify a cube root, the goal is to find factors under the radical that are perfect cubes\u00a0so that you can take their cube\u00a0root. We no longer need to be concerned about whether we have identified the principal root since we are now finding cube roots. Focus on finding identical trios\u00a0of factors as you simplify.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\sqrt[3]{40{{m}^{5}}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q617053\">Show Solution<\/span><\/p>\n<div id=\"q617053\" class=\"hidden-answer\" style=\"display: none\">Factor 40 into prime factors.<\/p>\n<p>[latex]\\sqrt[3]{5\\cdot 2\\cdot 2\\cdot 2\\cdot {{m}^{5}}}[\/latex]<\/p>\n<p>Since you are looking for the cube root, you need to find factors that appear 3 times under the radical. Rewrite [latex]2\\cdot 2\\cdot 2[\/latex] as [latex]{{2}^{3}}[\/latex].<\/p>\n<p>[latex]\\sqrt[3]{{{2}^{3}}\\cdot 5\\cdot {{m}^{5}}}[\/latex]<\/p>\n<p>Rewrite [latex]{{m}^{5}}[\/latex] as [latex]{{m}^{3}}\\cdot {{m}^{2}}[\/latex].<\/p>\n<p>[latex]\\sqrt[3]{{{2}^{3}}\\cdot 5\\cdot {{m}^{3}}\\cdot {{m}^{2}}}[\/latex]<\/p>\n<p>Rewrite the expression as a product of multiple radicals.<\/p>\n<p>[latex]\\sqrt[3]{{{2}^{3}}}\\cdot \\sqrt[3]{5}\\cdot \\sqrt[3]{{{m}^{3}}}\\cdot \\sqrt[3]{{{m}^{2}}}[\/latex]<\/p>\n<p>Simplify and multiply.<\/p>\n<p>[latex]2\\cdot \\sqrt[3]{5}\\cdot m\\cdot \\sqrt[3]{{{m}^{2}}}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\sqrt[3]{40{{m}^{5}}}=2m\\sqrt[3]{5{{m}^{2}}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Remember that you can take the cube root of a negative expression. In the next example we will simplify a cube root with a negative radicand.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\sqrt[3]{-27{{x}^{4}}{{y}^{3}}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q670300\">Show Solution<\/span><\/p>\n<div id=\"q670300\" class=\"hidden-answer\" style=\"display: none\">Factor the expression into cubes.<\/p>\n<p>Separate the cubed factors into individual radicals.<\/p>\n<p>[latex]\\begin{array}{r}\\sqrt[3]{-1\\cdot 27\\cdot {{x}^{4}}\\cdot {{y}^{3}}}\\\\\\sqrt[3]{{{(-1)}^{3}}\\cdot {{(3)}^{3}}\\cdot {{x}^{3}}\\cdot x\\cdot {{y}^{3}}}\\\\\\sqrt[3]{{{(-1)}^{3}}}\\cdot \\sqrt[3]{{{(3)}^{3}}}\\cdot \\sqrt[3]{{{x}^{3}}}\\cdot \\sqrt[3]{x}\\cdot \\sqrt[3]{{{y}^{3}}}\\end{array}[\/latex]<\/p>\n<p>Simplify the cube roots.<\/p>\n<p>[latex]-1\\cdot 3\\cdot x\\cdot y\\cdot \\sqrt[3]{x}[\/latex]<\/p>\n<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>You could check your answer by performing the inverse operation. If you are right, when you cube [latex]-3xy\\sqrt[3]{x}[\/latex] you should get [latex]-27{{x}^{4}}{{y}^{3}}[\/latex].<\/p>\n<p>[latex]\\begin{array}{l}\\left( -3xy\\sqrt[3]{x} \\right)\\left( -3xy\\sqrt[3]{x} \\right)\\left( -3xy\\sqrt[3]{x} \\right)\\\\-3\\cdot -3\\cdot -3\\cdot x\\cdot x\\cdot x\\cdot y\\cdot y\\cdot y\\cdot \\sqrt[3]{x}\\cdot \\sqrt[3]{x}\\cdot \\sqrt[3]{x}\\\\-27\\cdot {{x}^{3}}\\cdot {{y}^{3}}\\cdot \\sqrt[3]{{{x}^{3}}}\\\\-27{{x}^{3}}{{y}^{3}}\\cdot x\\\\-27{{x}^{4}}{{y}^{3}}\\end{array}[\/latex]<\/p>\n<p>You can also skip the step of factoring out the negative one once you are comfortable with identifying cubes.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\sqrt[3]{-24{{a}^{5}}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q473861\">Show Solution<\/span><\/p>\n<div id=\"q473861\" class=\"hidden-answer\" style=\"display: none\">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>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 following video we show more examples of simlifying cube roots.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-8\" title=\"Simplify Cube Roots (Not Perfect Cube Radicands)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/BtJruOpmHCE?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Simplifying fourth roots<\/h2>\n<p>Now let&#8217;s move to simplifying fourth degree roots. \u00a0No matter what root you are simplifying, the same idea applies, find cubes for cube roots, powers of four for fourth roots, etc. Recall that when your simplified expression contains an even indexed radical and a variable factor with an odd exponent, you need to apply an absolute value.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\sqrt[4]{81{{x}^{8}}{{y}^{3}}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q295348\">Show Solution<\/span><\/p>\n<div id=\"q295348\" class=\"hidden-answer\" style=\"display: none\">Rewrite the expression.<\/p>\n<p>[latex]\\sqrt[4]{81}\\cdot \\sqrt[4]{{{x}^{8}}}\\cdot \\sqrt[4]{{{y}^{3}}}[\/latex]<\/p>\n<p>Factor each radicand.<\/p>\n<p>[latex]\\sqrt[4]{3\\cdot 3\\cdot 3\\cdot 3}\\cdot \\sqrt[4]{{{x}^{2}}\\cdot {{x}^{2}}\\cdot {{x}^{2}}\\cdot {{x}^{2}}}\\cdot \\sqrt[4]{{{y}^{3}}}[\/latex]<\/p>\n<p>Simplify.<\/p>\n<p>[latex]\\begin{array}{r}\\sqrt[4]{{{3}^{4}}}\\cdot \\sqrt[4]{{{({{x}^{2}})}^{4}}}\\cdot \\sqrt[4]{{{y}^{3}}}\\\\3\\cdot {{x}^{2}}\\cdot \\sqrt[4]{{{y}^{3}}}\\end{array}[\/latex]<\/p>\n<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>An alternative method to factoring is to rewrite the expression with rational exponents, then use the rules of exponents to simplify. \u00a0You may find that you prefer one method over the other. Either way, it is nice to have options. We will show the last example again, using this idea.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\sqrt[4]{81{{x}^{8}}{{y}^{3}}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q324337\">Show Solution<\/span><\/p>\n<div id=\"q324337\" class=\"hidden-answer\" style=\"display: none\">Rewrite the radical using rational exponents.<\/p>\n<p>[latex]{{(81{{x}^{8}}{{y}^{3}})}^{\\frac{1}{4}}}[\/latex]<\/p>\n<p>Use the rules of exponents to simplify the expression.<\/p>\n<p>[latex]\\begin{array}{r}{{81}^{\\frac{1}{4}}}\\cdot {{x}^{\\frac{8}{4}}}\\cdot {{y}^{\\frac{3}{4}}}\\\\{{(3\\cdot 3\\cdot 3\\cdot 3)}^{\\frac{1}{4}}}{{x}^{2}}{{y}^{\\frac{3}{4}}}\\\\{{({{3}^{4}})}^{\\frac{1}{4}}}{{x}^{2}}{{y}^{\\frac{3}{4}}}\\\\3{{x}^{2}}{{y}^{\\frac{3}{4}}}\\end{array}[\/latex]<\/p>\n<p>Change the expression with the rational exponent back to radical form.<\/p>\n<p>[latex]3{{x}^{2}}\\sqrt[4]{{{y}^{3}}}[\/latex]<\/p>\n<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>In the following video we show another example of how to simplify a fourth and fifth root.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-9\" title=\"Simplify Nth Roots with Variables\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/op2LEb0YRyw?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>For our last example, we will simplify\u00a0a more complicated expression, [latex]\\large\\frac{10{{b}^{2}}{{c}^{2}}}{c\\sqrt[3]{8{{b}^{4}}}}[\/latex]<i>.<\/i><i> <\/i>This expression has two variables, a fraction, and a radical. Let\u2019s take it step-by-step and see if using fractional exponents can help us simplify it.<br \/>\nWe will\u00a0start by simplifying the denominator, since this is where the radical sign is located. Recall that an exponent in the denominator or a fraction can be rewritten as a negative exponent.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\large\\frac{10{{b}^{2}}{{c}^{2}}}{c\\sqrt[3]{8{{b}^{4}}}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q962386\">Show Solution<\/span><\/p>\n<div id=\"q962386\" class=\"hidden-answer\" style=\"display: none\">Separate the factors in the denominator.<\/p>\n<p>[latex]\\frac{10{{b}^{2}}{{c}^{2}}}{c\\cdot \\sqrt[3]{8}\\cdot \\sqrt[3]{{{b}^{4}}}}[\/latex]<\/p>\n<p>Take the cube root of 8, which is 2.<\/p>\n<p>[latex]\\frac{10{{b}^{2}}{{c}^{2}}}{c\\cdot 2\\cdot \\sqrt[3]{{{b}^{4}}}}[\/latex]<\/p>\n<p>Rewrite the radical using a fractional exponent.<\/p>\n<p>[latex]\\frac{10{{b}^{2}}{{c}^{2}}}{c\\cdot 2\\cdot {{b}^{\\frac{4}{3}}}}[\/latex]<\/p>\n<p>Rewrite the fraction as a series of factors in order to cancel factors (see next step).<\/p>\n<p>[latex]\\frac{10}{2}\\cdot \\frac{{{c}^{2}}}{c}\\cdot \\frac{{{b}^{2}}}{{{b}^{\\frac{4}{3}}}}[\/latex]<\/p>\n<p>Simplify the constant and <i>c<\/i> factors.<\/p>\n<p>[latex]5\\cdot c\\cdot \\frac{{{b}^{2}}}{{{b}^{\\frac{4}{3}}}}[\/latex]<\/p>\n<p>Use the rule of negative exponents,\u00a0<i>n<\/i><sup>&#8211;<\/sup><i><sup>x<\/sup><\/i><i>=<\/i>[latex]\\frac{1}{{{n}^{x}}}[\/latex], to rewrite [latex]\\frac{1}{{{b}^{\\tfrac{4}{3}}}}[\/latex] as [latex]{{b}^{-\\tfrac{4}{3}}}[\/latex].<\/p>\n<p>[latex]5c{{b}^{2}}{{b}^{-\\ \\frac{4}{3}}}[\/latex]<\/p>\n<p>Combine the <i>b<\/i> factors by adding the exponents.<\/p>\n<p>[latex]5c{{b}^{\\frac{2}{3}}}[\/latex]<\/p>\n<p>Change the expression with the fractional exponent back to radical form. By convention, an expression is not usually considered simplified if it has a fractional exponent or a radical in the denominator.<\/p>\n<p>[latex]5c\\sqrt[3]{{{b}^{2}}}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\frac{10{{b}^{2}}{{c}^{2}}}{c\\sqrt[3]{8{{b}^{4}}}}=5c\\sqrt[3]{{{b}^{2}}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Well, that took a while, but you did it. You applied what you know about fractional exponents, negative exponents, and the rules of exponents to simplify the expression.<\/p>\n<p>In our last video we show how to use rational exponents to simplify radical expressions.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-10\" title=\"Simplify Radicals Using Rational Exponents\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/CfxhFRHUq_M?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Summary<\/h2>\n<p>A radical expression is a mathematical way of representing the <i>n<\/i>th root of a number. Square roots and cube roots are the most common radicals, but a root can be any number. To simplify radical expressions, look for exponential factors within the radical, and then use the property [latex]\\sqrt[n]{{{x}^{n}}}=x[\/latex] if <i>n<\/i> is odd, and [latex]\\sqrt[n]{{{x}^{n}}}=\\left| x \\right|[\/latex] if <i>n<\/i> is even to pull out quantities. All rules of integer operations and exponents apply when simplifying radical expressions.<\/p>\n<p>The steps to consider when simplifying a radical are outlined below.<\/p>\n<div class=\"textbox shaded\">\n<h3>Simplifying a radical<\/h3>\n<p>When working with exponents and radicals:<\/p>\n<ul>\n<li>If <i>n<\/i> is odd, [latex]\\sqrt[n]{{{x}^{n}}}=x[\/latex].<\/li>\n<li>If <i>n<\/i> is even, [latex]\\sqrt[n]{{{x}^{n}}}=\\left| x \\right|[\/latex]. (The absolute value accounts for the fact that if <i>x<\/i> is negative and raised to an even power, that number will be positive, as will the <i>n<\/i>th principal root of that number.)<\/li>\n<\/ul>\n<\/div>\n<h2>Summary<\/h2>\n<p>The square root of a number is the number which, when multiplied by itself, gives the original number. Principal square roots are always positive and the square root of 0 is 0. You can only take the square root of values that are greater than or equal to 0. The square root of a perfect square will be an integer. Other roots can be simplified by identifying factors that are perfect squares, cubes, etc. Nth roots can be approximated using trial and error or a calculator.<\/p>\n<p>Any radical in the form [latex]\\sqrt[n]{a^{x}}[\/latex]\u00a0 can be written using a fractional exponent in the form [latex]a^{\\frac{x}{n}}[\/latex]. Rewriting radicals using fractional exponents can be useful in simplifying some radical expressions. When working with fractional exponents, remember that fractional exponents are subject to all of the same rules as other exponents when they appear in algebraic expressions.<\/p>\n<h2><\/h2>\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-1646\">\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 a Variety of Square Expressions (Simplify Perfectly). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/2cWAkmJoaDQ\">https:\/\/youtu.be\/2cWAkmJoaDQ<\/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 Perfect Nth Roots. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/vA2DkcUSRSk\">https:\/\/youtu.be\/vA2DkcUSRSk<\/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>Approximate a Square Root to Two Decimal Places Using Trial and Error. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/iNfalyW7olk\">https:\/\/youtu.be\/iNfalyW7olk<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>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 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 Square Roots with Variables. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/q7LqsKPoAKo\">https:\/\/youtu.be\/q7LqsKPoAKo<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Simplify Cube Roots (Not Perfect Cube Radicands). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/BtJruOpmHCE\">https:\/\/youtu.be\/BtJruOpmHCE<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Simplify Nth Roots with Variables. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/op2LEb0YRyw\">https:\/\/youtu.be\/op2LEb0YRyw<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Simplify Radicals Using Rational Exponents. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/CfxhFRHUq_M\">https:\/\/youtu.be\/CfxhFRHUq_M<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Unit 16: Radical Expressions and Quadratic Equations, from Developmental Math: An Open Program. <strong>Provided by<\/strong>: Monterey Institute of Technology. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/nrocnetwork.org\/dm-opentext\">http:\/\/nrocnetwork.org\/dm-opentext<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Precalculus. <strong>Authored by<\/strong>: Abramson, Jay. <strong>Provided by<\/strong>: Open Stax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":21,"menu_order":2,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Simplify a Variety of Square Expressions (Simplify Perfectly)\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/2cWAkmJoaDQ\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Simplify 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