{"id":188,"date":"2023-11-08T16:10:19","date_gmt":"2023-11-08T16:10:19","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/chapter\/read-or-watch-rational-exponents\/"},"modified":"2026-02-05T11:20:57","modified_gmt":"2026-02-05T11:20:57","slug":"5-2-rational-exponents","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/chapter\/5-2-rational-exponents\/","title":{"raw":"5.2 Rational Exponents","rendered":"5.2 Rational Exponents"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Rewrite expressions with rational exponents using radical notation.<\/li>\r\n \t<li>Rewrite radical expressions using rational exponents.<\/li>\r\n \t<li>Rewrite expressions containing negative exponents using positive rational exponents.<\/li>\r\n \t<li>Simplify expressions with rational exponents using the product and\/or quotient rules.<\/li>\r\n \t<li>Simplify expressions with negative rational exponents using the product and\/or quotient rules.<\/li>\r\n \t<li>Simplify expressions with rational exponents using the power rule.<\/li>\r\n \t<li>Simplify an expression involving a product and\/or quotient raised to a rational exponent.<\/li>\r\n \t<li>Simplify radical expressions by rewriting using rational exponents.<\/li>\r\n<\/ul>\r\n<\/div>\r\nWhen we write a radical expression like\u00a0[latex] \\sqrt{3}[\/latex], what we mean is that\u00a0[latex] \\sqrt{3}[\/latex] is the number such that squaring it results in 3. In other words,\r\n<p style=\"text-align: center;\">[latex]\\left(\\sqrt{3}\\right)^2 = 3 [\/latex]<\/p>\r\nHowever, something similar can happen when applying the Power Rule of exponents, [latex]\\left(a^m\\right)^n = a^{m\\cdot n}[\/latex]:\r\n<p style=\"text-align: center;\">[latex]\\left(3^{\\frac{1}{2}}\\right)^2 =3^{\\frac{1}{2}\\cdot 2} = 3^1 = 3 [\/latex]<\/p>\r\nAs long as it is legal to use the exponent properties with <em>rational<\/em> exponents, what we have shown is that [latex]\\sqrt{3}=3^{\\frac{1}{2}}[\/latex], since squaring both of them results in the same number! This suggests a possible interpretation of a [latex]\\dfrac{1}{2}[\/latex] exponent:\r\n<div class=\"textbox key-takeaways\">\r\n<h3>RATIONAL EXPONENTS<\/h3>\r\nWe define [latex]a^{\\frac{1}{2}} = \\sqrt{a}[\/latex].\r\n\r\nUsing similar logic as above, we can similarly define [latex]a^{\\frac{1}{n}} = \\sqrt[n]{a}[\/latex] for any positive integer [latex]n \\geq 2[\/latex] since raising both sides to the\u00a0[latex]n[\/latex]th power results in\u00a0[latex]a[\/latex]. If [latex]a[\/latex] is negative and [latex]n[\/latex] is even, these expressions are not real numbers.\r\n\r\n<\/div>\r\nWe can restate the above box as <strong>\"the denominator of the exponent is the index of the radical.\"<\/strong>\r\n\r\nHaving difficulty imagining a number being raised to a rational power? 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<h2>Write an Expression with a Rational Exponent as a Radical<\/h2>\r\nRadicals and rational exponents are alternate ways of expressing the same thing. \u00a0In the table below, we show equivalent ways to express radicals: with a radical, with a rational exponent, and as a principal root.\r\n<table style=\"width: 30%; height: 168px;\">\r\n<thead>\r\n<tr style=\"height: 42px;\">\r\n<th style=\"height: 42px;\">\r\n<p style=\"text-align: center;\">Radical Form<\/p>\r\n<\/th>\r\n<th style=\"height: 42px;\">\r\n<p style=\"text-align: center;\">Exponent Form<\/p>\r\n<\/th>\r\n<th style=\"height: 42px;\">\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 style=\"height: 42px;\">\r\n<td style=\"text-align: center; height: 42px;\">[latex] \\sqrt{16}[\/latex]<\/td>\r\n<td style=\"text-align: center; height: 42px;\">[latex] {{16}^{\\tfrac{1}{2}}}[\/latex]<\/td>\r\n<td style=\"text-align: center; height: 42px;\">[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 42px;\">\r\n<td style=\"text-align: center; height: 42px;\">[latex] \\sqrt{25}[\/latex]<\/td>\r\n<td style=\"text-align: center; height: 42px;\">[latex] {{25}^{\\tfrac{1}{2}}}[\/latex]<\/td>\r\n<td style=\"text-align: center; height: 42px;\">[latex]5[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 42px;\">\r\n<td style=\"text-align: center; height: 42px;\">[latex] \\sqrt{100}[\/latex]<\/td>\r\n<td style=\"text-align: center; height: 42px;\">[latex] {{100}^{\\tfrac{1}{2}}}[\/latex]<\/td>\r\n<td style=\"text-align: center; height: 42px;\">[latex]10[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nLet us 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\u00a0[latex]3[\/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<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;\">[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">[latex] \\sqrt[3]{125}[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex] {{125}^{\\tfrac{1}{3}}}[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]5[\/latex]<\/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;\">[latex]10[\/latex]<\/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\nWe can write radicals with rational exponents, and as we will see when we simplify more complex radical expressions, this can make things easier. A big advantage is that rewriting radicals as rational exponents could potentially allow us to use our exponent properties to simplify them.\r\n<h2>Rational Exponents Whose Numerator is Not Equal to One<\/h2>\r\nAll of the numerators for the rational exponents in the examples above were\u00a0[latex]1[\/latex]. Suppose we wanted to make sense of [latex]3^{\\frac{2}{3}}[\/latex]. We can again use the Power Rule of exponents (in two different ways!) to rewrite this expression:\r\n<p style=\"text-align: center;\">[latex] \\begin{align} =&amp;3^{^{\\frac{2}{3}}}=3^{\\frac{1}{3}\\cdot 2}=\\left(3^{\\frac{1}{3}}\\right)^2=(\\sqrt[3]{3})^2 \\textsf{ ... OR ...}\\\\ =&amp;3^{\\frac{2}{3}}=3^{2\\cdot \\frac{1}{3}}=\\left(3^{2}\\right)^{\\frac{1}{3}}=\\sqrt[3]{3^2}\\end{align}[\/latex]<\/p>\r\nThis leads us to our main rule for this section:\r\n<div class=\"textbox key-takeaways\">\r\n<h3>GENERAL RATIONAL EXPONENTS<\/h3>\r\nThe two forms [latex]\\sqrt[n]{a^{m}}[\/latex] and [latex]a^{\\frac{m}{n}}[\/latex] are equivalent.\u00a0 Both of these are also equivalent to\u00a0[latex](\\sqrt[n]{a})^m[\/latex] as long as\u00a0[latex]\\sqrt[n]{a}[\/latex] exists.\r\n\r\n<\/div>\r\nThis rule generalizes and includes the previous rule, since if [latex]m=1[\/latex] then the rule says the same thing.\r\n\r\nHere are some examples of using our new rule:\r\n<table style=\"width: 30%;\">\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=\"aligncenter wp-image-3198\" 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=\"Equation: fifth root of 7 squared equals 7 to the two-fifths power. Arrow points to 2 exponent and is labeled 'radicand'. Second arrow points to denominator 5 in exponent and is labeled 'root\/index'.\" width=\"380\" height=\"227\" \/>\r\n\r\nTo rewrite a radical using a rational exponent, the power to which the radicand is raised becomes the numerator and the index becomes the denominator.\r\n\r\nLet's practice this concept with a few examples.\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{1}{7}}[\/latex]<\/li>\r\n \t<li>[latex]{5}^{\\frac{4}{7}}[\/latex]<\/li>\r\n \t<li>[latex]{9}^{\\frac{3}{2}}[\/latex] (Also simplify the result)<\/li>\r\n \t<li>[latex] {{(2x)}^{^{\\frac{1}{3}}}}[\/latex]<\/li>\r\n \t<li>[latex] 2{{x}^{^{\\frac{1}{3}}}}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"200228\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"200228\"]\r\n<ol>\r\n \t<li>[latex]{x}^{\\frac{1}{7}}[\/latex], the numerator is\u00a0[latex]1[\/latex] and the denominator is\u00a0[latex]7[\/latex], therefore we will have the seventh root of [latex]x[\/latex], [latex]\\sqrt[7]{x}.[\/latex]<\/li>\r\n \t<li>[latex]{5}^{\\frac{4}{7}}[\/latex], the numerator is\u00a0[latex]4[\/latex] and the denominator is\u00a0[latex]7[\/latex], so we will have the seventh root of\u00a0[latex]5[\/latex] raised to the fourth power. [latex]\\sqrt[7]{5^4}.[\/latex]<\/li>\r\n \t<li>[latex]{9}^{\\frac{3}{2}}[\/latex], the numerator is\u00a0[latex]3[\/latex] and the denominator is\u00a0[latex]2[\/latex], so we will have the square root of\u00a0[latex]9[\/latex] raised to the fourth power, [latex]\\sqrt{9^3}.[\/latex] Note that this is still tricky to evaluate without a calculator, so it is easier in this case to use the alternative form of the radical with the power on the outside,\u00a0[latex]\\left(\\sqrt{9}\\right)^3.[\/latex] Now simplify,\u00a0[latex]\\left(\\sqrt{9}\\right)^3=3^3=27.[\/latex]<\/li>\r\n \t<li>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.<\/li>\r\n \t<li>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 [latex]x[\/latex]<i>, <\/i>but not the\u00a0[latex]2[\/latex].<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\nNotice in the last two examples that use of parentheses can change how the exponent is applied. Remember from working with polynomials that [latex]2x^3[\/latex] means only to cube the\u00a0[latex]x[\/latex] and not the coefficient\u00a0[latex]2.[\/latex]\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nRewrite each expression using a rational exponent.\r\n<ol>\r\n \t<li>[latex] \\sqrt[4]{81}[\/latex]<\/li>\r\n \t<li>[latex] 4\\sqrt[3]{xy}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt[6]{{{a}^{3}}}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt[12]{16^3}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"612743\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"612743\"]\r\n<ol>\r\n \t<li><span style=\"font-size: 1rem; text-align: initial;\">The fourth root can be rewritten as the exponent [latex] \\dfrac{1}{4}[\/latex]. Remove the radical and place the exponent next to the base.\u00a0 <\/span>\r\n<p style=\"text-align: center;\"><span style=\"font-size: 1rem; text-align: initial;\">[latex]\\sqrt[4]{81}=81^{1\/4}[\/latex]<\/span><\/p>\r\n<\/li>\r\n \t<li>Rewrite the radical using a rational exponent. The index determines the denominator. In this case, the index of the radical is\u00a0[latex]3[\/latex], so the rational exponent will be [latex] \\dfrac{1}{3}[\/latex].\r\n<p style=\"text-align: center;\">[latex] 4\\sqrt[3]{xy}=4{{(xy)}^{1\/3}}[\/latex]<\/p>\r\nSince\u00a0[latex]4[\/latex] is outside the radical, it is not included in the grouping symbol and the exponent does not refer to it.<\/li>\r\n \t<li>[latex]\\sqrt[n]{a^{m}}[\/latex] can be rewritten as\u00a0[latex]a^{\\frac{m}{n}}[\/latex], so in this case [latex]n=6,\\text{ and }m=3[\/latex], therefore\r\n<p style=\"text-align: center;\">[latex]\\sqrt[6]{{{a}^{3}}}={{a}^{3\/6}}={{a}^{1\/2}}[\/latex]<\/p>\r\nNote that this final answer is equivalent to [latex]\\sqrt{a}.[\/latex] This process is sometimes called <strong>reducing the index<\/strong>.<\/li>\r\n \t<li>[latex]\\sqrt[n]{a^{m}}[\/latex] can be rewritten as\u00a0[latex]a^{\\frac{m}{n}}[\/latex], so in this case [latex]n=12,\\text{ and }m=3[\/latex], therefore\r\n<p style=\"text-align: center;\">[latex]\\sqrt[12]{16^3}={16}^{3\/12}={16}^{1\/4}[\/latex]<\/p>\r\nThis is equivalent to\u00a0[latex]\\sqrt[4]{16},[\/latex] which evaluates as\u00a0[latex]2.[\/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>Using Exponent Properties<\/h2>\r\nWe have implied a few times that we would like our exponent properties to work in general for all rational numbers, and not just for integers like they did in <a href=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/chapter\/read-properties-of-exponents-and-intro-to-polynomials\/\">Section 3.1<\/a>.\u00a0Fortunately, this will be the case and all of our rules work consistently with radicals if we define it this way. Let's state this very important fact here.\r\n<div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>EXPONENT PROPERTIES<\/h3>\r\nAll of our exponent properties work for rational exponents (before, we only said they were true for integers).\r\n\r\nIn addition, since radicals are equivalent to rational exponents, this means that\r\n<p style=\"text-align: center;\"><strong>Whenever we want to investigate or simplify radicals, <\/strong>\r\n<strong>we often convert to exponents and use exponent properties!<\/strong><\/p>\r\nWe will make use of this philosophy in the coming sections.\r\n\r\n<\/div>\r\nFor now, here is a reminder of our previous exponent properties (which now work for rational exponents):\r\n<table style=\"width: 70%; height: 322px;\"><caption style=\"caption-side: top; font-size: large;\">Properties of Exponents<\/caption>\r\n<thead>\r\n<tr style=\"height: 14px;\">\r\n<th style=\"height: 14px;\">\u00a0Name<\/th>\r\n<th style=\"height: 14px;\">Property<\/th>\r\n<th style=\"height: 14px;\">Example<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr style=\"height: 42px;\">\r\n<td style=\"height: 42px;\">Product Rule<\/td>\r\n<td style=\"height: 42px;\">[latex]{\\large a^{m}\\cdot{a}^{n}=a^{m+n}}[\/latex]<\/td>\r\n<td style=\"height: 42px;\">[latex]{\\large x^{1\/3}\\cdot{x}^{1\/2}=x^{1\/3+1\/2}=x^{5\/6}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 28px;\">\r\n<td style=\"height: 28px;\">Quotient Rule<\/td>\r\n<td style=\"height: 28px;\">[latex]{\\large \\dfrac{a^{m}}{{a}^{n}}=a^{m-n}}[\/latex]<\/td>\r\n<td style=\"height: 28px;\">[latex]{\\large \\dfrac{x^{1\/2}}{{x}^{1\/3}}=x^{1\/2-1\/3}=x^{1\/6}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 42px;\">\r\n<td style=\"height: 42px;\">Power Rule<\/td>\r\n<td style=\"height: 42px;\">[latex]{\\large \\left(a^{m}\\right)^{n}=a^{m\\cdot{n}}}[\/latex]<\/td>\r\n<td style=\"height: 42px;\">[latex]{\\large \\left(x^{1\/5}\\right)^{2}=x^{2\\cdot{1\/5}}=x^{2\/5}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 56px;\">\r\n<td style=\"height: 56px;\">Power of a Product Rule<\/td>\r\n<td style=\"height: 56px;\">[latex]{\\large \\left(ab\\right)^{n}=a^{n}\\cdot b^{n}}[\/latex]<\/td>\r\n<td style=\"height: 56px;\">[latex]{\\large \\left(2y\\right)^{1\/2}=2^{1\/2}\\cdot y^{1\/2}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 56px;\">\r\n<td style=\"height: 56px;\">Power of a Quotient Rule<\/td>\r\n<td style=\"height: 56px;\">[latex]{\\large \\left(\\dfrac{a}{b}\\right)^{n}=\\dfrac{{a}^{n}}{{b}^{n}}}[\/latex]<\/td>\r\n<td style=\"height: 56px;\">[latex]{\\large \\left(\\dfrac{x}{3}\\right)^{1\/3}=\\dfrac{{x}^{1\/3}}{{3}^{1\/3}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 42px;\">\r\n<td style=\"height: 42px;\">Negative Exponent Rule<\/td>\r\n<td style=\"height: 42px;\">[latex]{\\large {a}^{-n}=\\dfrac{1}{{a}^{n}}}[\/latex]<\/td>\r\n<td style=\"height: 42px;\">[latex]{\\large {x}^{-1\/2}=\\dfrac{1}{{x}^{1\/2}}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h2><span style=\"font-size: 1rem; font-weight: normal; orphans: 1; text-align: initial; color: #373d3f;\">Here are a few examples of using our exponent properties to simplify a rational expression with many factors. This is similar to what we've done before with integer exponents, but now we use rational exponents. Such an expression is considered fully simplified when three things are done.<\/span><\/h2>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Simplifying requirements<\/h3>\r\nTo simplify a rational expression involving factors with exponents, you must use Properties of Exponents to do the following (in any order):\r\n<ul>\r\n \t<li>Apply all powers, for example [latex]\\left(x^2y\\right)^3=\\left(x^2\\right)^3y^3=x^6y^3[\/latex]<\/li>\r\n \t<li>Combine so each variable appears only once<\/li>\r\n \t<li>Rewrite all negative exponents in an equivalent form with positive exponents<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSimplify each expression. Assume all variables represent positive quantities and write your final answer using no negative exponents.\r\n<ol>\r\n \t<li>[latex]\\left(\\frac{x^{2\/5}}{y^{4\/3}}\\right)^{-1\/3}[\/latex]<\/li>\r\n \t<li>[latex]\\frac{x^{1\/5}y^{1\/2}(x^{1\/2}y^{-1})^{-7\/5}}{x^{3\/2}y^{8\/5}}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"899415\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"899415\"]\r\n\r\n1. There are many ways to approach problems of simplifying exponents. Each approach is correct as long as you apply valid exponent rules at each step. In this example, we begin with using quotient to a power:\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\left(\\frac{x^{2\/5}}{y^{4\/3}}\\right)^{-1\/3}=&amp;\\quad\\frac{(x^{2\/5})^{-1\/3}}{(y^{4\/3})^{-1\/3}}&amp;&amp;\\color{blue}{\\textsf{Power of a Quotient}}\\\\=&amp;\\quad\\frac{x^{\\frac{2}{5}\\cdot -\\frac{1}{3}}}{y^{\\frac{4}{3}\\cdot-\\frac{1}{3}}}&amp;&amp;\\color{blue}{\\textsf{Power Rule}}\\\\=&amp;\\quad\\frac{x^{-2\/15}}{y^{-4\/9}}\\\\=&amp;\\quad\\frac{y^{4\/9}}{x^{2\/15}}\\end{align}[\/latex]<\/p>\r\nIn the last step we handled the negative exponents by switching them to the opposite side of the fraction bar.\r\n\r\n2. Again, there are different ways we can simplify the expression. We start this example by applying the power in the numerator.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\r\n\\frac{x^{1\/5}y^{1\/2}(x^{1\/2}y^{-1})^{-7\/5}}{x^{3\/2}y^{8\/5}}=&amp;\\quad\\frac{x^{1\/5}y^{1\/2}(x^{1\/2}y^{-1})^{-7\/5}}{x^{3\/2}y^{8\/5}}\\\\\r\n=&amp;\\quad\\frac{x^{1\/5}y^{1\/2}(x^{1\/2})^{-7\/5}(y^{-1})^{-7\/5}}{x^{3\/2}y^{8\/5}}&amp;&amp;\\color{blue}{\\textsf{Power of a Product}}\\\\\r\n=&amp;\\quad\\frac{x^{1\/5}y^{1\/2}x^{-7\/10}y^{7\/5}}{x^{3\/2}y^{8\/5}}&amp;&amp;\\color{blue}{\\textsf{Power Rule}}\\\\\r\n=&amp;\\quad\\frac{x^{\\frac{1}{5}-\\frac{7}{10}}y^{\\frac{1}{2}+\\frac{7}{5}}}{x^{3\/2}y^{8\/5}}&amp;&amp;\\color{blue}{\\textsf{Product Rule}}\\\\\r\n=&amp;\\quad\\frac{x^{-5\/10}y^{\\frac{19}{10}}}{x^{3\/2}y^{8\/5}}\\\\\r\n=&amp;\\quad x^{-\\frac{1}{2}-\\frac{3}{2}}y^{\\frac{19}{10}-\\frac{8}{5}}&amp;&amp;\\color{blue}{\\textsf{Quotient Rule}}\\\\\r\n=&amp;\\quad x^{-\\frac{4}{2}}y^{\\frac{3}{10}}\\\\\r\n=&amp;\\quad \\frac{y^{\\frac{3}{10}}}{x^2}&amp;&amp;\\color{blue}{\\textsf{Negative Exponent Rule}}\\\\\r\n\\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the next example, we take advantage of exponent properties to rewrite and simplify an expression with multiple radicals. This is a preview of the next few sections, where we will use exponent properties frequently to work with radicals.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSimplify. Write your answer in radical notation. [latex]\\sqrt[3]{\\sqrt[4]{x}}[\/latex]\r\n\r\n[reveal-answer q=\"899515\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"899515\"]\r\n\r\nWe first convert both radicals to exponent notation, and then use properties of exponents to simplify.\r\n<p style=\"text-align: center;\">[latex]\\sqrt[3]{\\sqrt[4]{x}}=\\sqrt[3]{x^{1\/4}}=\\left(x^{1\/4}\\right)^{1\/3}=x^{\\frac{1}{4}\\cdot\\frac{1}{3}}=x^{1\/12}[\/latex]<\/p>\r\nNow, convert back to radical notation, [latex]\\sqrt[12]{x}.[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Summary<\/h2>\r\nAny radical in the form [latex]\\sqrt[n]{a^{x}}[\/latex] can be written using a rational exponent in the form [latex]a^{\\frac{x}{n}}[\/latex]. Rewriting radicals using rational exponents can be useful when simplifying some radical expressions. When working with rational exponents, remember that rational exponents are subject to all of the same rules as other exponents when they appear in algebraic expressions.","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Rewrite expressions with rational exponents using radical notation.<\/li>\n<li>Rewrite radical expressions using rational exponents.<\/li>\n<li>Rewrite expressions containing negative exponents using positive rational exponents.<\/li>\n<li>Simplify expressions with rational exponents using the product and\/or quotient rules.<\/li>\n<li>Simplify expressions with negative rational exponents using the product and\/or quotient rules.<\/li>\n<li>Simplify expressions with rational exponents using the power rule.<\/li>\n<li>Simplify an expression involving a product and\/or quotient raised to a rational exponent.<\/li>\n<li>Simplify radical expressions by rewriting using rational exponents.<\/li>\n<\/ul>\n<\/div>\n<p>When we write a radical expression like\u00a0[latex]\\sqrt{3}[\/latex], what we mean is that\u00a0[latex]\\sqrt{3}[\/latex] is the number such that squaring it results in 3. In other words,<\/p>\n<p style=\"text-align: center;\">[latex]\\left(\\sqrt{3}\\right)^2 = 3[\/latex]<\/p>\n<p>However, something similar can happen when applying the Power Rule of exponents, [latex]\\left(a^m\\right)^n = a^{m\\cdot n}[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]\\left(3^{\\frac{1}{2}}\\right)^2 =3^{\\frac{1}{2}\\cdot 2} = 3^1 = 3[\/latex]<\/p>\n<p>As long as it is legal to use the exponent properties with <em>rational<\/em> exponents, what we have shown is that [latex]\\sqrt{3}=3^{\\frac{1}{2}}[\/latex], since squaring both of them results in the same number! This suggests a possible interpretation of a [latex]\\dfrac{1}{2}[\/latex] exponent:<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>RATIONAL EXPONENTS<\/h3>\n<p>We define [latex]a^{\\frac{1}{2}} = \\sqrt{a}[\/latex].<\/p>\n<p>Using similar logic as above, we can similarly define [latex]a^{\\frac{1}{n}} = \\sqrt[n]{a}[\/latex] for any positive integer [latex]n \\geq 2[\/latex] since raising both sides to the\u00a0[latex]n[\/latex]th power results in\u00a0[latex]a[\/latex]. If [latex]a[\/latex] is negative and [latex]n[\/latex] is even, these expressions are not real numbers.<\/p>\n<\/div>\n<p>We can restate the above box as <strong>&#8220;the denominator of the exponent is the index of the radical.&#8221;<\/strong><\/p>\n<p>Having difficulty imagining a number being raised to a rational power? 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<h2>Write an Expression with a Rational Exponent as a Radical<\/h2>\n<p>Radicals and rational exponents are alternate ways of expressing the same thing. \u00a0In the table below, we show equivalent ways to express radicals: with a radical, with a rational exponent, and as a principal root.<\/p>\n<table style=\"width: 30%; height: 168px;\">\n<thead>\n<tr style=\"height: 42px;\">\n<th style=\"height: 42px;\">\n<p style=\"text-align: center;\">Radical Form<\/p>\n<\/th>\n<th style=\"height: 42px;\">\n<p style=\"text-align: center;\">Exponent Form<\/p>\n<\/th>\n<th style=\"height: 42px;\">\n<p style=\"text-align: center;\">Principal Root<\/p>\n<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 42px;\">\n<td style=\"text-align: center; height: 42px;\">[latex]\\sqrt{16}[\/latex]<\/td>\n<td style=\"text-align: center; height: 42px;\">[latex]{{16}^{\\tfrac{1}{2}}}[\/latex]<\/td>\n<td style=\"text-align: center; height: 42px;\">[latex]4[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 42px;\">\n<td style=\"text-align: center; height: 42px;\">[latex]\\sqrt{25}[\/latex]<\/td>\n<td style=\"text-align: center; height: 42px;\">[latex]{{25}^{\\tfrac{1}{2}}}[\/latex]<\/td>\n<td style=\"text-align: center; height: 42px;\">[latex]5[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 42px;\">\n<td style=\"text-align: center; height: 42px;\">[latex]\\sqrt{100}[\/latex]<\/td>\n<td style=\"text-align: center; height: 42px;\">[latex]{{100}^{\\tfrac{1}{2}}}[\/latex]<\/td>\n<td style=\"text-align: center; height: 42px;\">[latex]10[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Let us 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\u00a0[latex]3[\/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<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;\">[latex]2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]\\sqrt[3]{125}[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]{{125}^{\\tfrac{1}{3}}}[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]5[\/latex]<\/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;\">[latex]10[\/latex]<\/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>We can write radicals with rational exponents, and as we will see when we simplify more complex radical expressions, this can make things easier. A big advantage is that rewriting radicals as rational exponents could potentially allow us to use our exponent properties to simplify them.<\/p>\n<h2>Rational Exponents Whose Numerator is Not Equal to One<\/h2>\n<p>All of the numerators for the rational exponents in the examples above were\u00a0[latex]1[\/latex]. Suppose we wanted to make sense of [latex]3^{\\frac{2}{3}}[\/latex]. We can again use the Power Rule of exponents (in two different ways!) to rewrite this expression:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align} =&3^{^{\\frac{2}{3}}}=3^{\\frac{1}{3}\\cdot 2}=\\left(3^{\\frac{1}{3}}\\right)^2=(\\sqrt[3]{3})^2 \\textsf{ ... OR ...}\\\\ =&3^{\\frac{2}{3}}=3^{2\\cdot \\frac{1}{3}}=\\left(3^{2}\\right)^{\\frac{1}{3}}=\\sqrt[3]{3^2}\\end{align}[\/latex]<\/p>\n<p>This leads us to our main rule for this section:<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>GENERAL RATIONAL EXPONENTS<\/h3>\n<p>The two forms [latex]\\sqrt[n]{a^{m}}[\/latex] and [latex]a^{\\frac{m}{n}}[\/latex] are equivalent.\u00a0 Both of these are also equivalent to\u00a0[latex](\\sqrt[n]{a})^m[\/latex] as long as\u00a0[latex]\\sqrt[n]{a}[\/latex] exists.<\/p>\n<\/div>\n<p>This rule generalizes and includes the previous rule, since if [latex]m=1[\/latex] then the rule says the same thing.<\/p>\n<p>Here are some examples of using our new rule:<\/p>\n<table style=\"width: 30%;\">\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=\"aligncenter wp-image-3198\" 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=\"Equation: fifth root of 7 squared equals 7 to the two-fifths power. Arrow points to 2 exponent and is labeled 'radicand'. Second arrow points to denominator 5 in exponent and is labeled 'root\/index'.\" width=\"380\" height=\"227\" \/><\/p>\n<p>To rewrite a radical using a rational exponent, the power to which the radicand is raised becomes the numerator and the index becomes the denominator.<\/p>\n<p>Let&#8217;s practice this concept with a few examples.<\/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{1}{7}}[\/latex]<\/li>\n<li>[latex]{5}^{\\frac{4}{7}}[\/latex]<\/li>\n<li>[latex]{9}^{\\frac{3}{2}}[\/latex] (Also simplify the result)<\/li>\n<li>[latex]{{(2x)}^{^{\\frac{1}{3}}}}[\/latex]<\/li>\n<li>[latex]2{{x}^{^{\\frac{1}{3}}}}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q200228\">Show Solution<\/span><\/p>\n<div id=\"q200228\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]{x}^{\\frac{1}{7}}[\/latex], the numerator is\u00a0[latex]1[\/latex] and the denominator is\u00a0[latex]7[\/latex], therefore we will have the seventh root of [latex]x[\/latex], [latex]\\sqrt[7]{x}.[\/latex]<\/li>\n<li>[latex]{5}^{\\frac{4}{7}}[\/latex], the numerator is\u00a0[latex]4[\/latex] and the denominator is\u00a0[latex]7[\/latex], so we will have the seventh root of\u00a0[latex]5[\/latex] raised to the fourth power. [latex]\\sqrt[7]{5^4}.[\/latex]<\/li>\n<li>[latex]{9}^{\\frac{3}{2}}[\/latex], the numerator is\u00a0[latex]3[\/latex] and the denominator is\u00a0[latex]2[\/latex], so we will have the square root of\u00a0[latex]9[\/latex] raised to the fourth power, [latex]\\sqrt{9^3}.[\/latex] Note that this is still tricky to evaluate without a calculator, so it is easier in this case to use the alternative form of the radical with the power on the outside,\u00a0[latex]\\left(\\sqrt{9}\\right)^3.[\/latex] Now simplify,\u00a0[latex]\\left(\\sqrt{9}\\right)^3=3^3=27.[\/latex]<\/li>\n<li>Rewrite the expression with the fractional exponent as a radical. The denominator of the fraction determines the root, in this case the cube root.\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.<\/li>\n<li>Rewrite the expression with the fractional exponent as a radical. The denominator of the fraction determines the root, in this case the cube root.\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 [latex]x[\/latex]<i>, <\/i>but not the\u00a0[latex]2[\/latex].<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<p>Notice in the last two examples that use of parentheses can change how the exponent is applied. Remember from working with polynomials that [latex]2x^3[\/latex] means only to cube the\u00a0[latex]x[\/latex] and not the coefficient\u00a0[latex]2.[\/latex]<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Rewrite each expression using a rational exponent.<\/p>\n<ol>\n<li>[latex]\\sqrt[4]{81}[\/latex]<\/li>\n<li>[latex]4\\sqrt[3]{xy}[\/latex]<\/li>\n<li>[latex]\\sqrt[6]{{{a}^{3}}}[\/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=\"q612743\">Show Solution<\/span><\/p>\n<div id=\"q612743\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li><span style=\"font-size: 1rem; text-align: initial;\">The fourth root can be rewritten as the exponent [latex]\\dfrac{1}{4}[\/latex]. Remove the radical and place the exponent next to the base.\u00a0 <\/span>\n<p style=\"text-align: center;\"><span style=\"font-size: 1rem; text-align: initial;\">[latex]\\sqrt[4]{81}=81^{1\/4}[\/latex]<\/span><\/p>\n<\/li>\n<li>Rewrite the radical using a rational exponent. The index determines the denominator. In this case, the index of the radical is\u00a0[latex]3[\/latex], so the rational exponent will be [latex]\\dfrac{1}{3}[\/latex].\n<p style=\"text-align: center;\">[latex]4\\sqrt[3]{xy}=4{{(xy)}^{1\/3}}[\/latex]<\/p>\n<p>Since\u00a0[latex]4[\/latex] is outside the radical, it is not included in the grouping symbol and the exponent does not refer to it.<\/li>\n<li>[latex]\\sqrt[n]{a^{m}}[\/latex] can be rewritten as\u00a0[latex]a^{\\frac{m}{n}}[\/latex], so in this case [latex]n=6,\\text{ and }m=3[\/latex], therefore\n<p style=\"text-align: center;\">[latex]\\sqrt[6]{{{a}^{3}}}={{a}^{3\/6}}={{a}^{1\/2}}[\/latex]<\/p>\n<p>Note that this final answer is equivalent to [latex]\\sqrt{a}.[\/latex] This process is sometimes called <strong>reducing the index<\/strong>.<\/li>\n<li>[latex]\\sqrt[n]{a^{m}}[\/latex] can be rewritten as\u00a0[latex]a^{\\frac{m}{n}}[\/latex], so in this case [latex]n=12,\\text{ and }m=3[\/latex], therefore\n<p style=\"text-align: center;\">[latex]\\sqrt[12]{16^3}={16}^{3\/12}={16}^{1\/4}[\/latex]<\/p>\n<p>This is equivalent to\u00a0[latex]\\sqrt[4]{16},[\/latex] which evaluates as\u00a0[latex]2.[\/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-1\" 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>Using Exponent Properties<\/h2>\n<p>We have implied a few times that we would like our exponent properties to work in general for all rational numbers, and not just for integers like they did in <a href=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/chapter\/read-properties-of-exponents-and-intro-to-polynomials\/\">Section 3.1<\/a>.\u00a0Fortunately, this will be the case and all of our rules work consistently with radicals if we define it this way. Let&#8217;s state this very important fact here.<\/p>\n<div>\n<div class=\"textbox key-takeaways\">\n<h3>EXPONENT PROPERTIES<\/h3>\n<p>All of our exponent properties work for rational exponents (before, we only said they were true for integers).<\/p>\n<p>In addition, since radicals are equivalent to rational exponents, this means that<\/p>\n<p style=\"text-align: center;\"><strong>Whenever we want to investigate or simplify radicals, <\/strong><br \/>\n<strong>we often convert to exponents and use exponent properties!<\/strong><\/p>\n<p>We will make use of this philosophy in the coming sections.<\/p>\n<\/div>\n<p>For now, here is a reminder of our previous exponent properties (which now work for rational exponents):<\/p>\n<table style=\"width: 70%; height: 322px;\">\n<caption style=\"caption-side: top; font-size: large;\">Properties of Exponents<\/caption>\n<thead>\n<tr style=\"height: 14px;\">\n<th style=\"height: 14px;\">\u00a0Name<\/th>\n<th style=\"height: 14px;\">Property<\/th>\n<th style=\"height: 14px;\">Example<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 42px;\">\n<td style=\"height: 42px;\">Product Rule<\/td>\n<td style=\"height: 42px;\">[latex]{\\large a^{m}\\cdot{a}^{n}=a^{m+n}}[\/latex]<\/td>\n<td style=\"height: 42px;\">[latex]{\\large x^{1\/3}\\cdot{x}^{1\/2}=x^{1\/3+1\/2}=x^{5\/6}}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 28px;\">\n<td style=\"height: 28px;\">Quotient Rule<\/td>\n<td style=\"height: 28px;\">[latex]{\\large \\dfrac{a^{m}}{{a}^{n}}=a^{m-n}}[\/latex]<\/td>\n<td style=\"height: 28px;\">[latex]{\\large \\dfrac{x^{1\/2}}{{x}^{1\/3}}=x^{1\/2-1\/3}=x^{1\/6}}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 42px;\">\n<td style=\"height: 42px;\">Power Rule<\/td>\n<td style=\"height: 42px;\">[latex]{\\large \\left(a^{m}\\right)^{n}=a^{m\\cdot{n}}}[\/latex]<\/td>\n<td style=\"height: 42px;\">[latex]{\\large \\left(x^{1\/5}\\right)^{2}=x^{2\\cdot{1\/5}}=x^{2\/5}}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 56px;\">\n<td style=\"height: 56px;\">Power of a Product Rule<\/td>\n<td style=\"height: 56px;\">[latex]{\\large \\left(ab\\right)^{n}=a^{n}\\cdot b^{n}}[\/latex]<\/td>\n<td style=\"height: 56px;\">[latex]{\\large \\left(2y\\right)^{1\/2}=2^{1\/2}\\cdot y^{1\/2}}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 56px;\">\n<td style=\"height: 56px;\">Power of a Quotient Rule<\/td>\n<td style=\"height: 56px;\">[latex]{\\large \\left(\\dfrac{a}{b}\\right)^{n}=\\dfrac{{a}^{n}}{{b}^{n}}}[\/latex]<\/td>\n<td style=\"height: 56px;\">[latex]{\\large \\left(\\dfrac{x}{3}\\right)^{1\/3}=\\dfrac{{x}^{1\/3}}{{3}^{1\/3}}}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 42px;\">\n<td style=\"height: 42px;\">Negative Exponent Rule<\/td>\n<td style=\"height: 42px;\">[latex]{\\large {a}^{-n}=\\dfrac{1}{{a}^{n}}}[\/latex]<\/td>\n<td style=\"height: 42px;\">[latex]{\\large {x}^{-1\/2}=\\dfrac{1}{{x}^{1\/2}}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2><span style=\"font-size: 1rem; font-weight: normal; orphans: 1; text-align: initial; color: #373d3f;\">Here are a few examples of using our exponent properties to simplify a rational expression with many factors. This is similar to what we&#8217;ve done before with integer exponents, but now we use rational exponents. Such an expression is considered fully simplified when three things are done.<\/span><\/h2>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Simplifying requirements<\/h3>\n<p>To simplify a rational expression involving factors with exponents, you must use Properties of Exponents to do the following (in any order):<\/p>\n<ul>\n<li>Apply all powers, for example [latex]\\left(x^2y\\right)^3=\\left(x^2\\right)^3y^3=x^6y^3[\/latex]<\/li>\n<li>Combine so each variable appears only once<\/li>\n<li>Rewrite all negative exponents in an equivalent form with positive exponents<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Simplify each expression. Assume all variables represent positive quantities and write your final answer using no negative exponents.<\/p>\n<ol>\n<li>[latex]\\left(\\frac{x^{2\/5}}{y^{4\/3}}\\right)^{-1\/3}[\/latex]<\/li>\n<li>[latex]\\frac{x^{1\/5}y^{1\/2}(x^{1\/2}y^{-1})^{-7\/5}}{x^{3\/2}y^{8\/5}}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q899415\">Show Solution<\/span><\/p>\n<div id=\"q899415\" class=\"hidden-answer\" style=\"display: none\">\n<p>1. There are many ways to approach problems of simplifying exponents. Each approach is correct as long as you apply valid exponent rules at each step. In this example, we begin with using quotient to a power:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\left(\\frac{x^{2\/5}}{y^{4\/3}}\\right)^{-1\/3}=&\\quad\\frac{(x^{2\/5})^{-1\/3}}{(y^{4\/3})^{-1\/3}}&&\\color{blue}{\\textsf{Power of a Quotient}}\\\\=&\\quad\\frac{x^{\\frac{2}{5}\\cdot -\\frac{1}{3}}}{y^{\\frac{4}{3}\\cdot-\\frac{1}{3}}}&&\\color{blue}{\\textsf{Power Rule}}\\\\=&\\quad\\frac{x^{-2\/15}}{y^{-4\/9}}\\\\=&\\quad\\frac{y^{4\/9}}{x^{2\/15}}\\end{align}[\/latex]<\/p>\n<p>In the last step we handled the negative exponents by switching them to the opposite side of the fraction bar.<\/p>\n<p>2. Again, there are different ways we can simplify the expression. We start this example by applying the power in the numerator.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}  \\frac{x^{1\/5}y^{1\/2}(x^{1\/2}y^{-1})^{-7\/5}}{x^{3\/2}y^{8\/5}}=&\\quad\\frac{x^{1\/5}y^{1\/2}(x^{1\/2}y^{-1})^{-7\/5}}{x^{3\/2}y^{8\/5}}\\\\  =&\\quad\\frac{x^{1\/5}y^{1\/2}(x^{1\/2})^{-7\/5}(y^{-1})^{-7\/5}}{x^{3\/2}y^{8\/5}}&&\\color{blue}{\\textsf{Power of a Product}}\\\\  =&\\quad\\frac{x^{1\/5}y^{1\/2}x^{-7\/10}y^{7\/5}}{x^{3\/2}y^{8\/5}}&&\\color{blue}{\\textsf{Power Rule}}\\\\  =&\\quad\\frac{x^{\\frac{1}{5}-\\frac{7}{10}}y^{\\frac{1}{2}+\\frac{7}{5}}}{x^{3\/2}y^{8\/5}}&&\\color{blue}{\\textsf{Product Rule}}\\\\  =&\\quad\\frac{x^{-5\/10}y^{\\frac{19}{10}}}{x^{3\/2}y^{8\/5}}\\\\  =&\\quad x^{-\\frac{1}{2}-\\frac{3}{2}}y^{\\frac{19}{10}-\\frac{8}{5}}&&\\color{blue}{\\textsf{Quotient Rule}}\\\\  =&\\quad x^{-\\frac{4}{2}}y^{\\frac{3}{10}}\\\\  =&\\quad \\frac{y^{\\frac{3}{10}}}{x^2}&&\\color{blue}{\\textsf{Negative Exponent Rule}}\\\\  \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the next example, we take advantage of exponent properties to rewrite and simplify an expression with multiple radicals. This is a preview of the next few sections, where we will use exponent properties frequently to work with radicals.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Simplify. Write your answer in radical notation. [latex]\\sqrt[3]{\\sqrt[4]{x}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q899515\">Show Solution<\/span><\/p>\n<div id=\"q899515\" class=\"hidden-answer\" style=\"display: none\">\n<p>We first convert both radicals to exponent notation, and then use properties of exponents to simplify.<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt[3]{\\sqrt[4]{x}}=\\sqrt[3]{x^{1\/4}}=\\left(x^{1\/4}\\right)^{1\/3}=x^{\\frac{1}{4}\\cdot\\frac{1}{3}}=x^{1\/12}[\/latex]<\/p>\n<p>Now, convert back to radical notation, [latex]\\sqrt[12]{x}.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Summary<\/h2>\n<p>Any radical in the form [latex]\\sqrt[n]{a^{x}}[\/latex] can be written using a rational exponent in the form [latex]a^{\\frac{x}{n}}[\/latex]. Rewriting radicals using rational exponents can be useful when simplifying some radical expressions. When working with rational exponents, remember that rational exponents are subject to all of the same rules as other exponents when they appear in algebraic expressions.<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-188\">\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>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>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Unit 16: Radical Expressions and Quadratic Equations, from Developmental Math: An Open Program. <strong>Provided by<\/strong>: Monterey Institute of Technology. <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>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/pdm\">Public Domain: No Known Copyright<\/a><\/em>. <strong>License Terms<\/strong>: Dwonload fro 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":395986,"menu_order":2,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Write Expressions Using Radicals and Rational Exponents\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/5cWkVrANBWA\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Unit 16: Radical Expressions and Quadratic Equations, from Developmental Math: An Open Program\",\"author\":\"\",\"organization\":\"Monterey Institute of Technology\",\"url\":\"http:\/\/nrocnetwork.org\/dm-opentext\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Precalculus\",\"author\":\"Abramson, Jay\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"pd\",\"license_terms\":\"Dwonload fro free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\"}]","CANDELA_OUTCOMES_GUID":"5be86d34-aef9-412f-bd4a-b7dec7fdd046","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-188","chapter","type-chapter","status-publish","hentry"],"part":184,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/pressbooks\/v2\/chapters\/188","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/wp\/v2\/users\/395986"}],"version-history":[{"count":22,"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/pressbooks\/v2\/chapters\/188\/revisions"}],"predecessor-version":[{"id":2130,"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/pressbooks\/v2\/chapters\/188\/revisions\/2130"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/pressbooks\/v2\/parts\/184"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/pressbooks\/v2\/chapters\/188\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/wp\/v2\/media?parent=188"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=188"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/wp\/v2\/contributor?post=188"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/wp\/v2\/license?post=188"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}