{"id":187,"date":"2023-11-08T16:10:18","date_gmt":"2023-11-08T16:10:18","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/chapter\/1646\/"},"modified":"2024-07-29T15:20:56","modified_gmt":"2024-07-29T15:20:56","slug":"5-1-radical-functions-and-radical-expressions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/chapter\/5-1-radical-functions-and-radical-expressions\/","title":{"raw":"5.1 Radical Functions and Radical Expressions","rendered":"5.1 Radical Functions and Radical Expressions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Identify the radicand and the index of a radical expression.<\/li>\r\n \t<li>Simplify [latex]n[\/latex]th roots of expressions that are perfect [latex]n[\/latex]th powers.<\/li>\r\n \t<li>Evaluate [latex]n[\/latex]th root functions.<\/li>\r\n \t<li>Determine the domain of a radical function from its equation and write in interval notation.<\/li>\r\n<\/ul>\r\n<\/div>\r\nThe most common root is the <strong>square root<\/strong>. First, we will define what square roots are and how you find the square root of a number. Then we will apply similar ideas to define and evaluate [latex]n[\/latex]th roots.\r\n\r\nRoots are the inverse of exponents, much like multiplication is the inverse of division. When you are trying to find a square root of a number (say,\u00a0[latex]25[\/latex]), you are trying to find a number that can be multiplied by itself to create that original number. In the case of\u00a0[latex]25[\/latex], you find that [latex]5\\cdot5=25[\/latex], so\u00a0[latex]5[\/latex] is a square root.\r\n<h2>Square Roots<\/h2>\r\nThe symbol for square root is called a <strong>radical symbol<\/strong> and looks like this: [latex]\\sqrt{\\,\\,\\,}[\/latex]. The expression [latex] \\sqrt{25}[\/latex] is read \u201csquare 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.\" \/>\r\n\r\nThe following table shows a few more examples of evaluating radicals.\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>\u201cSquare root of 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{\\dfrac{1}{100}}[\/latex]<\/td>\r\n<td>\u201cSquare root of one hundredth\u201d<\/td>\r\n<td>[latex] \\sqrt{\\dfrac{1}{100}}=\\sqrt{\\dfrac{1}{10}\\cdot \\dfrac{1}{10}}=\\dfrac{1}{10}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex] \\sqrt{0.16}[\/latex]<\/td>\r\n<td>\u201cSquare root of 16 hundredths\u201d<\/td>\r\n<td>[latex] \\sqrt{0.16}=\\sqrt{\\dfrac{16}{100}}=\\sqrt{\\dfrac{4}{10}\\cdot \\dfrac{4}{10}}=\\dfrac{4}{10}=\\dfrac{2}{5}=0.4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex] \\sqrt{2}[\/latex]<\/td>\r\n<td>\u201cSquare root of 2\u201d<\/td>\r\n<td>[latex] \\sqrt{2} \\approx 1.414...[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNote in the last example that there is no \"nice\" number that multiplies by itself to get [latex]2[\/latex]. [latex] \\sqrt{2} [\/latex] is an example of an <strong>irrational number<\/strong>, which means it cannot be written as a quotient of two integers. As a decimal number it continues forever without repetition. A calculator is used to approximate its value as [latex] 1.414 [\/latex], since [latex] (1.414)^2 = 1.999396 [\/latex], which is close to [latex]2[\/latex]. More decimal places can be used to get the result closer to [latex]2[\/latex].\r\n\r\nIn our first example we will show you how to evaluate square roots.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSimplify each expression.\r\n<ol>\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{\\dfrac{9}{49}}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{-9}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"419579\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"419579\"]\r\n<ol>\r\n \t<li>[latex]\\sqrt{16}=4[\/latex] because [latex]{4}^{2}=16[\/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]. Note that \"distributing\" the square root results in a different, incorrect answer: [latex]\\sqrt{25}+\\sqrt{144}=5+12=\\color{Red}{17}[\/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{\\dfrac{9}{49}} = -\\sqrt{\\dfrac{3}{7}\\cdot \\dfrac{3}{7}} = -\\dfrac{3}{7}[\/latex].<\/p>\r\n<\/li>\r\n \t<li>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 of [latex]9[\/latex], and [latex]3^2[\/latex] will also give the result [latex]9[\/latex]. We say that [latex]\\sqrt{-9}[\/latex] does not exist as a real number, and the same is true for the square root of any negative number.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe last example we showed leads to an important characteristic of square roots. You can only take the square root of numbers that are nonnegative.\r\n<div class=\"textbox shaded\">\r\n<h3>SQUARE ROOTS OF NEGATIVE NUMBERS<\/h3>\r\nWe say that the square root of a negative number does not exist as a real number.\r\n\r\n<\/div>\r\nWe will come back to this idea later when we discuss the domain of radical functions.\r\n<h2>Principal Roots<\/h2>\r\nConsider [latex] \\sqrt{25}[\/latex] again. You may realize that there is another number that, when multiplied by itself, also results in\u00a0[latex]25[\/latex]. 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\u00a0[latex]25[\/latex], only\u00a0[latex]5[\/latex] is the principal root. We say that [latex]\u22125[\/latex] is called the <strong>negative root<\/strong>. We will never evaluate a radical as its negative root unless specifically asked to.\r\n\r\nIn the following video, we present more examples of how to find a principal square root.\r\n\r\nhttps:\/\/youtu.be\/2cWAkmJoaDQ\r\n<h2>Nth 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. For example, since [latex]5^3=125[\/latex], we can say that [latex]\\sqrt[3]{125}=5[\/latex].\r\n\r\nThe [latex]n[\/latex]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\u00a0[latex]2[\/latex]. In the radical expression, [latex]n[\/latex] is called the <strong>index<\/strong> of the radical.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Definition:\u00a0<em>n<\/em>th Root<\/h3>\r\nIf [latex]a[\/latex] is a real number with at least one [latex]n[\/latex]th root, then the <strong><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 [latex]n[\/latex]th power, equals [latex]a[\/latex]. The <strong>index<\/strong> of the radical is [latex]n[\/latex]. A square root has index of [latex]2[\/latex], and the index number does not need to be written.\r\n\r\n<\/div>\r\nThere is one interesting fact about cube roots that is not true of square roots. 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! Here is a general fact about the existence of roots with negative radicands.\r\n<div class=\"textbox shaded\">\r\n<h3>NTH ROOTS OF NEGATIVE NUMBERS<\/h3>\r\nWe say that an even index root of a negative number does not exist as a real number.\r\n\r\nOdd index roots of negative numbers always exist as real numbers.\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[3]{216}[\/latex]<\/li>\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 Solution[\/reveal-answer]\r\n[hidden-answer a=\"140298\"]\r\n<ol>\r\n \t<li>Write [latex]216[\/latex] as a perfect cube, [latex]{\\left(6\\right)}^{3}=216[\/latex]. So, [latex]\\sqrt[3]{216}=6[\/latex]<\/li>\r\n \t<li>Write [latex]-32[\/latex] as a perfect fifth power, [latex]{\\left(-2\\right)}^{5}=-32[\/latex]. So, [latex]\\sqrt[5]{-32}[\/latex] = [latex]-2[\/latex]<\/li>\r\n \t<li>Write [latex]81[\/latex] as a perfect fourth power, [latex]\\sqrt[4]{81}=\\sqrt[4]{3\\cdot3\\cdot3\\cdot3}=\\sqrt[4]{3^4}=3[\/latex]<\/li>\r\n \t<li>Since we have an\u00a0[latex]8[\/latex]th root (which is even) with a negative number as the radicand, this root does not have a real number result.<\/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 [latex]n[\/latex]th roots.\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]{-27},\\ \\sqrt[5]{-32}[\/latex], and [latex] \\sqrt[7]{-2187}[\/latex], but the radicals [latex] \\sqrt{-100},\\ \\sqrt[4]{-16}[\/latex], and [latex] \\sqrt[6]{-1}[\/latex] are not real numbers. Let's now state this using function vocabulary.\r\n<h2>Radical Functions<\/h2>\r\nThe function [latex]f(x)=\\sqrt{x}[\/latex] is a radical function. We can evaluate it just like any other function by using a number or expression for the input. For example,\r\n<p style=\"text-align: center;\">[latex]f(\\color{green}{4}) = \\sqrt{\\color{green}{4}}=2[\/latex].<\/p>\r\nAs discussed earlier, we cannot use any negative number as an input in this function. However, every nonnegative number can be used as input (even if the radical must be approximated). Thus, we conclude that the domain of [latex]f[\/latex] is all nonnegative numbers, or in interval notation [latex] [0, \\infty) [\/latex]. We must also ensure a nonnegative radicand for any even index.\r\n\r\nFor the function [latex]f(x)=\\sqrt[3]{x}[\/latex], note that negative inputs are allowed. In fact, there is no real number which cannot be used as input. So the domain of [latex]f[\/latex] is all real numbers, or [latex] (-\\infty, \\infty) [\/latex]. The same will be true of any radical function with an odd index.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Domain of Radical Functions<\/h3>\r\n[latex]f(x)=\\sqrt[n]{x}[\/latex] has domain [latex] [0, \\infty) [\/latex] if [latex]n[\/latex] is even, and domain [latex] (-\\infty, \\infty) [\/latex] if [latex]n[\/latex] is odd. When [latex]n[\/latex] is even, we can solve for the domain by setting up an inequality to ensure the radicand is nonnegative: [latex]x \\geq 0.[\/latex]\r\n\r\n<\/div>\r\nAs an example of the last statement, consider the domain of [latex]f(x)=\\sqrt[4]{x-3}[\/latex]. We cannot say the domain is [latex][0, \\infty)[\/latex] because if we use [latex]0[\/latex] as input, the result would be [latex]f(0)=\\sqrt[4]{0-3}=\\sqrt[4]{-3}[\/latex], which is not a real number. The key is that the radicand, [latex]\\color{Green}{x-3}[\/latex], is the quantity that must be nonnegative. We can state this with an inequality and solve:\r\n<p style=\"text-align: center;\">[latex]\r\n\\begin{align}\r\n\\color{Green}{x-3} &amp; \\geq 0 \\\\\r\nx &amp;\\geq 3\r\n\\end{align}\r\n[\/latex]<\/p>\r\nSo the domain of this function is the interval [latex][3, \\infty)[\/latex].\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nComplete each of the following:\r\n<ol>\r\n \t<li>Evaluate [latex]f(x) = \\sqrt[3]{3x+3}[\/latex] at [latex]x=-10[\/latex]<\/li>\r\n \t<li>Determine the domain of [latex]f(x)=\\sqrt{3x+7}[\/latex]<\/li>\r\n \t<li>Determine the domain of [latex]f(x)=\\sqrt[6]{5-x}[\/latex]<\/li>\r\n \t<li>Determine the domain of [latex]f(x)=\\sqrt[3]{5-x}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"140299\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"140299\"]\r\n<ol>\r\n \t<li>Note that since the index is [latex]3,[\/latex] which is odd, all numbers are in the domain of\u00a0 [latex]f[\/latex]. Evaluate [latex]f(-10) = \\sqrt[3]{3\\cdot -10+3} = \\sqrt[3]{-27} = -3[\/latex].<\/li>\r\n \t<li>Since the index is [latex]2,[\/latex] which is even, we set up an inequality to ensure the radicand is nonnegative:\r\n[latex]\\begin{align} 3x+7&amp;\\geq 0 \\\\ 3x &amp;\\geq -7 \\\\ x&amp;\\geq -\\frac{7}{3} \\end{align}[\/latex]\r\nThe domain is [latex] \\left[-\\dfrac{7}{3}, \\infty\\right) [\/latex].<\/li>\r\n \t<li>Since the index is [latex]6,[\/latex] which is even, we set up an inequality to ensure the radicand is nonnegative:\r\n[latex]\\begin{align}5-x&amp;\\geq 0 \\\\ -x &amp;\\geq -5 \\\\ x &amp;\\leq 5 \\end{align}[\/latex].\r\nNote in the last step that we divide by [latex]-1[\/latex], so we must reverse the inequality sign. The domain is [latex] (-\\infty, 5] [\/latex].<\/li>\r\n \t<li>Since the index is [latex]3,[\/latex] which is odd, the domain is all real numbers or [latex] (-\\infty, \\infty) [\/latex].<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Evaluating Radicals with Variables<\/h2>\r\nConsider the expression [latex] \\sqrt{{{x}^{2}}}[\/latex]. This looks like it should be equal to [latex] x[\/latex], right? Test some values for [latex] x[\/latex]\u00a0and see what happens.\r\n\r\nIn the chart below, look along each row and determine whether the value of [latex]x[\/latex]\u00a0is the same as the value of [latex] \\sqrt{{{x}^{2}}}[\/latex]. Where are they equal? Where are they not equal?\r\n<table style=\"width: 40%;\">\r\n<thead>\r\n<tr style=\"height: 30px;\">\r\n<th style=\"height: 30px;\">[latex]x[\/latex]<\/th>\r\n<th style=\"height: 30px;\">[latex]x^{2}[\/latex]<\/th>\r\n<th style=\"height: 30px;\">[latex]\\sqrt{x^{2}}[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"height: 30px;\">[latex]\u22125[\/latex]<\/td>\r\n<td style=\"height: 30px;\">[latex]25[\/latex]<\/td>\r\n<td style=\"height: 30px;\">[latex]5[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"height: 30px;\">[latex]\u22122[\/latex]<\/td>\r\n<td style=\"height: 30px;\">[latex]4[\/latex]<\/td>\r\n<td style=\"height: 30px;\">[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\">[latex]0[\/latex]<\/td>\r\n<td style=\"height: 15px;\">[latex]0[\/latex]<\/td>\r\n<td style=\"height: 15px;\">[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\">[latex]6[\/latex]<\/td>\r\n<td style=\"height: 15px;\">[latex]36[\/latex]<\/td>\r\n<td style=\"height: 15px;\">[latex]6[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15.125px;\">\r\n<td style=\"height: 15.125px;\">[latex]10[\/latex]<\/td>\r\n<td style=\"height: 15.125px;\">[latex]100[\/latex]<\/td>\r\n<td style=\"height: 15.125px;\">[latex]10[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNotice that in cases where [latex]x[\/latex]\u00a0is a negative number, [latex]\\sqrt{x^{2}}\\neq{x}[\/latex]. The correct result of the square root was always positive since it was defined as the principal root. We can generally say that [latex]\\sqrt{x^{2}}=\\left|x\\right|[\/latex] because the absolute value will ensure the result represents the principal root.\r\n<div class=\"textbox shaded\">\r\n<h3>Simplifying Radicals with Variables<\/h3>\r\nWhen finding the square root of an expression that contains variables raised to an even power, remember that [latex]\\sqrt{x^{2}}=\\left|x\\right|[\/latex]. The absolute value signs are used for any even index.\r\nAbsolute values should not be used for odd index radicals. So [latex]\\sqrt[n]{x^{n}}=x[\/latex] if [latex]n[\/latex] is odd.\r\nIf we already know that [latex]x[\/latex] is nonnegative, then the absolute value signs are not necessary at all.\r\n\r\n<\/div>\r\nFor example, [latex]\\sqrt{9x^{2}}=\\sqrt{(3x)^2}=\\left|3x\\right|[\/latex], and [latex]\\sqrt{16{{x}^{2}}{{y}^{2}}}=\\sqrt{(4xy)^2}=\\left|4xy\\right|[\/latex].\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSimplify each of the following:\r\n<ol>\r\n \t<li>[latex]\\sqrt{64y^2}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{(2y+5)^2}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt[5]{32x^5y^5}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{x^{12}}[\/latex]<\/li>\r\n \t<li>[latex]-3x\\sqrt{4x^2}[\/latex], assuming\u00a0[latex]x[\/latex] is nonnegative<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"140290\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"140290\"]\r\n<ol>\r\n \t<li>[latex]\\sqrt{64y^2}=\\sqrt{(8y)^2}=\\left|8y\\right|[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{(2y+5)^2}=\\left|2y+5\\right|[\/latex]. Note that the absolute value bars must contain the entire expression which was being squared.<\/li>\r\n \t<li>[latex]\\sqrt[5]{32x^5y^5}=2xy[\/latex]. Note that since the index is odd, the absolute value should not be used.<\/li>\r\n \t<li>[latex]\\sqrt{x^{12}}=\\sqrt{x^6 \\cdot x^6}=\\left|x^6\\right|[\/latex]. Notice that even powers of [latex]x[\/latex] will always be positive anyway, so the absolute values are not required this time. We can write the answer as [latex]x^6.[\/latex]<\/li>\r\n \t<li>[latex]-3x\\sqrt{4x^2}=-3x\\sqrt{(2x)^2}=-3x \\cdot 2x=-6x^2[\/latex]. Similar to the previous example, absolute values are not needed because of the assumption. If there is an expression outside the radical, multiply after simplifying the radical.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Summary<\/h2>\r\nA square root of a number is a number which, when multiplied by itself, gives the original number. Principal square roots are always positive or zero. In general, an [latex]n[\/latex]th root is a number which, when raised to the [latex]n[\/latex]th power, gives the original number. You can only take an even root of numbers that are greater than or equal to\u00a0[latex]0[\/latex]. When simplifying radical expressions with variables, be careful to use absolute values when the index is even.","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Identify the radicand and the index of a radical expression.<\/li>\n<li>Simplify [latex]n[\/latex]th roots of expressions that are perfect [latex]n[\/latex]th powers.<\/li>\n<li>Evaluate [latex]n[\/latex]th root functions.<\/li>\n<li>Determine the domain of a radical function from its equation and write in interval notation.<\/li>\n<\/ul>\n<\/div>\n<p>The most common root is the <strong>square root<\/strong>. First, we will define what square roots are and how you find the square root of a number. Then we will apply similar ideas to define and evaluate [latex]n[\/latex]th roots.<\/p>\n<p>Roots are the inverse of exponents, much like multiplication is the inverse of division. When you are trying to find a square root of a number (say,\u00a0[latex]25[\/latex]), you are trying to find a number that can be multiplied by itself to create that original number. In the case of\u00a0[latex]25[\/latex], you find that [latex]5\\cdot5=25[\/latex], so\u00a0[latex]5[\/latex] is a square root.<\/p>\n<h2>Square Roots<\/h2>\n<p>The symbol for square root is called a <strong>radical symbol<\/strong> and looks like this: [latex]\\sqrt{\\,\\,\\,}[\/latex]. The expression [latex]\\sqrt{25}[\/latex] is read \u201csquare 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.\" \/><\/p>\n<p>The following table shows a few more examples of evaluating radicals.<\/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>\u201cSquare root of thirty-six\u201d<\/td>\n<td>[latex]\\sqrt{36}=\\sqrt{6\\cdot 6}=6[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\sqrt{\\dfrac{1}{100}}[\/latex]<\/td>\n<td>\u201cSquare root of one hundredth\u201d<\/td>\n<td>[latex]\\sqrt{\\dfrac{1}{100}}=\\sqrt{\\dfrac{1}{10}\\cdot \\dfrac{1}{10}}=\\dfrac{1}{10}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\sqrt{0.16}[\/latex]<\/td>\n<td>\u201cSquare root of 16 hundredths\u201d<\/td>\n<td>[latex]\\sqrt{0.16}=\\sqrt{\\dfrac{16}{100}}=\\sqrt{\\dfrac{4}{10}\\cdot \\dfrac{4}{10}}=\\dfrac{4}{10}=\\dfrac{2}{5}=0.4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\sqrt{2}[\/latex]<\/td>\n<td>\u201cSquare root of 2\u201d<\/td>\n<td>[latex]\\sqrt{2} \\approx 1.414...[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Note in the last example that there is no &#8220;nice&#8221; number that multiplies by itself to get [latex]2[\/latex]. [latex]\\sqrt{2}[\/latex] is an example of an <strong>irrational number<\/strong>, which means it cannot be written as a quotient of two integers. As a decimal number it continues forever without repetition. A calculator is used to approximate its value as [latex]1.414[\/latex], since [latex](1.414)^2 = 1.999396[\/latex], which is close to [latex]2[\/latex]. More decimal places can be used to get the result closer to [latex]2[\/latex].<\/p>\n<p>In our first example we will show you how to evaluate square roots.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Simplify each expression.<\/p>\n<ol>\n<li>[latex]\\sqrt{16}[\/latex]<\/li>\n<li>[latex]\\sqrt{25+144}[\/latex]<\/li>\n<li>[latex]-\\sqrt{\\dfrac{9}{49}}[\/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 Solution<\/span><\/p>\n<div id=\"q419579\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]\\sqrt{16}=4[\/latex] because [latex]{4}^{2}=16[\/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]. Note that &#8220;distributing&#8221; the square root results in a different, incorrect answer: [latex]\\sqrt{25}+\\sqrt{144}=5+12=\\color{Red}{17}[\/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{\\dfrac{9}{49}} = -\\sqrt{\\dfrac{3}{7}\\cdot \\dfrac{3}{7}} = -\\dfrac{3}{7}[\/latex].<\/p>\n<\/li>\n<li>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 of [latex]9[\/latex], and [latex]3^2[\/latex] will also give the result [latex]9[\/latex]. We say that [latex]\\sqrt{-9}[\/latex] does not exist as a real number, and the same is true for the square root of any negative number.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>The last example we showed leads to an important characteristic of square roots. You can only take the square root of numbers that are nonnegative.<\/p>\n<div class=\"textbox shaded\">\n<h3>SQUARE ROOTS OF NEGATIVE NUMBERS<\/h3>\n<p>We say that the square root of a negative number does not exist as a real number.<\/p>\n<\/div>\n<p>We will come back to this idea later when we discuss the domain of radical functions.<\/p>\n<h2>Principal Roots<\/h2>\n<p>Consider [latex]\\sqrt{25}[\/latex] again. You may realize that there is another number that, when multiplied by itself, also results in\u00a0[latex]25[\/latex]. 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\u00a0[latex]25[\/latex], only\u00a0[latex]5[\/latex] is the principal root. We say that [latex]\u22125[\/latex] is called the <strong>negative root<\/strong>. We will never evaluate a radical as its negative root unless specifically asked to.<\/p>\n<p>In the following video, we present more examples of how to find a principal 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<h2>Nth 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. For example, since [latex]5^3=125[\/latex], we can say that [latex]\\sqrt[3]{125}=5[\/latex].<\/p>\n<p>The [latex]n[\/latex]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\u00a0[latex]2[\/latex]. In the radical expression, [latex]n[\/latex] is called the <strong>index<\/strong> of the radical.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Definition:\u00a0<em>n<\/em>th Root<\/h3>\n<p>If [latex]a[\/latex] is a real number with at least one [latex]n[\/latex]th root, then the <strong><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 [latex]n[\/latex]th power, equals [latex]a[\/latex]. The <strong>index<\/strong> of the radical is [latex]n[\/latex]. A square root has index of [latex]2[\/latex], and the index number does not need to be written.<\/p>\n<\/div>\n<p>There is one interesting fact about cube roots that is not true of square roots. 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! Here is a general fact about the existence of roots with negative radicands.<\/p>\n<div class=\"textbox shaded\">\n<h3>NTH ROOTS OF NEGATIVE NUMBERS<\/h3>\n<p>We say that an even index root of a negative number does not exist as a real number.<\/p>\n<p>Odd index roots of negative numbers always exist as real numbers.<\/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[3]{216}[\/latex]<\/li>\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 Solution<\/span><\/p>\n<div id=\"q140298\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>Write [latex]216[\/latex] as a perfect cube, [latex]{\\left(6\\right)}^{3}=216[\/latex]. So, [latex]\\sqrt[3]{216}=6[\/latex]<\/li>\n<li>Write [latex]-32[\/latex] as a perfect fifth power, [latex]{\\left(-2\\right)}^{5}=-32[\/latex]. So, [latex]\\sqrt[5]{-32}[\/latex] = [latex]-2[\/latex]<\/li>\n<li>Write [latex]81[\/latex] as a perfect fourth power, [latex]\\sqrt[4]{81}=\\sqrt[4]{3\\cdot3\\cdot3\\cdot3}=\\sqrt[4]{3^4}=3[\/latex]<\/li>\n<li>Since we have an\u00a0[latex]8[\/latex]th root (which is even) with a negative number as the radicand, this root does not have a real number result.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video, we show more examples of how to evaluate [latex]n[\/latex]th roots.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" 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]{-27},\\ \\sqrt[5]{-32}[\/latex], and [latex]\\sqrt[7]{-2187}[\/latex], but the radicals [latex]\\sqrt{-100},\\ \\sqrt[4]{-16}[\/latex], and [latex]\\sqrt[6]{-1}[\/latex] are not real numbers. Let&#8217;s now state this using function vocabulary.<\/p>\n<h2>Radical Functions<\/h2>\n<p>The function [latex]f(x)=\\sqrt{x}[\/latex] is a radical function. We can evaluate it just like any other function by using a number or expression for the input. For example,<\/p>\n<p style=\"text-align: center;\">[latex]f(\\color{green}{4}) = \\sqrt{\\color{green}{4}}=2[\/latex].<\/p>\n<p>As discussed earlier, we cannot use any negative number as an input in this function. However, every nonnegative number can be used as input (even if the radical must be approximated). Thus, we conclude that the domain of [latex]f[\/latex] is all nonnegative numbers, or in interval notation [latex][0, \\infty)[\/latex]. We must also ensure a nonnegative radicand for any even index.<\/p>\n<p>For the function [latex]f(x)=\\sqrt[3]{x}[\/latex], note that negative inputs are allowed. In fact, there is no real number which cannot be used as input. So the domain of [latex]f[\/latex] is all real numbers, or [latex](-\\infty, \\infty)[\/latex]. The same will be true of any radical function with an odd index.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Domain of Radical Functions<\/h3>\n<p>[latex]f(x)=\\sqrt[n]{x}[\/latex] has domain [latex][0, \\infty)[\/latex] if [latex]n[\/latex] is even, and domain [latex](-\\infty, \\infty)[\/latex] if [latex]n[\/latex] is odd. When [latex]n[\/latex] is even, we can solve for the domain by setting up an inequality to ensure the radicand is nonnegative: [latex]x \\geq 0.[\/latex]<\/p>\n<\/div>\n<p>As an example of the last statement, consider the domain of [latex]f(x)=\\sqrt[4]{x-3}[\/latex]. We cannot say the domain is [latex][0, \\infty)[\/latex] because if we use [latex]0[\/latex] as input, the result would be [latex]f(0)=\\sqrt[4]{0-3}=\\sqrt[4]{-3}[\/latex], which is not a real number. The key is that the radicand, [latex]\\color{Green}{x-3}[\/latex], is the quantity that must be nonnegative. We can state this with an inequality and solve:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}  \\color{Green}{x-3} & \\geq 0 \\\\  x &\\geq 3  \\end{align}[\/latex]<\/p>\n<p>So the domain of this function is the interval [latex][3, \\infty)[\/latex].<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Complete each of the following:<\/p>\n<ol>\n<li>Evaluate [latex]f(x) = \\sqrt[3]{3x+3}[\/latex] at [latex]x=-10[\/latex]<\/li>\n<li>Determine the domain of [latex]f(x)=\\sqrt{3x+7}[\/latex]<\/li>\n<li>Determine the domain of [latex]f(x)=\\sqrt[6]{5-x}[\/latex]<\/li>\n<li>Determine the domain of [latex]f(x)=\\sqrt[3]{5-x}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q140299\">Show Solution<\/span><\/p>\n<div id=\"q140299\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>Note that since the index is [latex]3,[\/latex] which is odd, all numbers are in the domain of\u00a0 [latex]f[\/latex]. Evaluate [latex]f(-10) = \\sqrt[3]{3\\cdot -10+3} = \\sqrt[3]{-27} = -3[\/latex].<\/li>\n<li>Since the index is [latex]2,[\/latex] which is even, we set up an inequality to ensure the radicand is nonnegative:<br \/>\n[latex]\\begin{align} 3x+7&\\geq 0 \\\\ 3x &\\geq -7 \\\\ x&\\geq -\\frac{7}{3} \\end{align}[\/latex]<br \/>\nThe domain is [latex]\\left[-\\dfrac{7}{3}, \\infty\\right)[\/latex].<\/li>\n<li>Since the index is [latex]6,[\/latex] which is even, we set up an inequality to ensure the radicand is nonnegative:<br \/>\n[latex]\\begin{align}5-x&\\geq 0 \\\\ -x &\\geq -5 \\\\ x &\\leq 5 \\end{align}[\/latex].<br \/>\nNote in the last step that we divide by [latex]-1[\/latex], so we must reverse the inequality sign. The domain is [latex](-\\infty, 5][\/latex].<\/li>\n<li>Since the index is [latex]3,[\/latex] which is odd, the domain is all real numbers or [latex](-\\infty, \\infty)[\/latex].<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<h2>Evaluating Radicals with Variables<\/h2>\n<p>Consider the expression [latex]\\sqrt{{{x}^{2}}}[\/latex]. This looks like it should be equal to [latex]x[\/latex], right? Test some values for [latex]x[\/latex]\u00a0and see what happens.<\/p>\n<p>In the chart below, look along each row and determine whether the value of [latex]x[\/latex]\u00a0is the same as the value of [latex]\\sqrt{{{x}^{2}}}[\/latex]. Where are they equal? Where are they not equal?<\/p>\n<table style=\"width: 40%;\">\n<thead>\n<tr style=\"height: 30px;\">\n<th style=\"height: 30px;\">[latex]x[\/latex]<\/th>\n<th style=\"height: 30px;\">[latex]x^{2}[\/latex]<\/th>\n<th style=\"height: 30px;\">[latex]\\sqrt{x^{2}}[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 30px;\">\n<td style=\"height: 30px;\">[latex]\u22125[\/latex]<\/td>\n<td style=\"height: 30px;\">[latex]25[\/latex]<\/td>\n<td style=\"height: 30px;\">[latex]5[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 30px;\">\n<td style=\"height: 30px;\">[latex]\u22122[\/latex]<\/td>\n<td style=\"height: 30px;\">[latex]4[\/latex]<\/td>\n<td style=\"height: 30px;\">[latex]2[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\">[latex]0[\/latex]<\/td>\n<td style=\"height: 15px;\">[latex]0[\/latex]<\/td>\n<td style=\"height: 15px;\">[latex]0[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\">[latex]6[\/latex]<\/td>\n<td style=\"height: 15px;\">[latex]36[\/latex]<\/td>\n<td style=\"height: 15px;\">[latex]6[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15.125px;\">\n<td style=\"height: 15.125px;\">[latex]10[\/latex]<\/td>\n<td style=\"height: 15.125px;\">[latex]100[\/latex]<\/td>\n<td style=\"height: 15.125px;\">[latex]10[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Notice that in cases where [latex]x[\/latex]\u00a0is a negative number, [latex]\\sqrt{x^{2}}\\neq{x}[\/latex]. The correct result of the square root was always positive since it was defined as the principal root. We can generally say that [latex]\\sqrt{x^{2}}=\\left|x\\right|[\/latex] because the absolute value will ensure the result represents the principal root.<\/p>\n<div class=\"textbox shaded\">\n<h3>Simplifying Radicals with Variables<\/h3>\n<p>When finding the square root of an expression that contains variables raised to an even power, remember that [latex]\\sqrt{x^{2}}=\\left|x\\right|[\/latex]. The absolute value signs are used for any even index.<br \/>\nAbsolute values should not be used for odd index radicals. So [latex]\\sqrt[n]{x^{n}}=x[\/latex] if [latex]n[\/latex] is odd.<br \/>\nIf we already know that [latex]x[\/latex] is nonnegative, then the absolute value signs are not necessary at all.<\/p>\n<\/div>\n<p>For example, [latex]\\sqrt{9x^{2}}=\\sqrt{(3x)^2}=\\left|3x\\right|[\/latex], and [latex]\\sqrt{16{{x}^{2}}{{y}^{2}}}=\\sqrt{(4xy)^2}=\\left|4xy\\right|[\/latex].<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Simplify each of the following:<\/p>\n<ol>\n<li>[latex]\\sqrt{64y^2}[\/latex]<\/li>\n<li>[latex]\\sqrt{(2y+5)^2}[\/latex]<\/li>\n<li>[latex]\\sqrt[5]{32x^5y^5}[\/latex]<\/li>\n<li>[latex]\\sqrt{x^{12}}[\/latex]<\/li>\n<li>[latex]-3x\\sqrt{4x^2}[\/latex], assuming\u00a0[latex]x[\/latex] is nonnegative<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q140290\">Show Solution<\/span><\/p>\n<div id=\"q140290\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]\\sqrt{64y^2}=\\sqrt{(8y)^2}=\\left|8y\\right|[\/latex]<\/li>\n<li>[latex]\\sqrt{(2y+5)^2}=\\left|2y+5\\right|[\/latex]. Note that the absolute value bars must contain the entire expression which was being squared.<\/li>\n<li>[latex]\\sqrt[5]{32x^5y^5}=2xy[\/latex]. Note that since the index is odd, the absolute value should not be used.<\/li>\n<li>[latex]\\sqrt{x^{12}}=\\sqrt{x^6 \\cdot x^6}=\\left|x^6\\right|[\/latex]. Notice that even powers of [latex]x[\/latex] will always be positive anyway, so the absolute values are not required this time. We can write the answer as [latex]x^6.[\/latex]<\/li>\n<li>[latex]-3x\\sqrt{4x^2}=-3x\\sqrt{(2x)^2}=-3x \\cdot 2x=-6x^2[\/latex]. Similar to the previous example, absolute values are not needed because of the assumption. If there is an expression outside the radical, multiply after simplifying the radical.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<h2>Summary<\/h2>\n<p>A square root of a number is a number which, when multiplied by itself, gives the original number. Principal square roots are always positive or zero. In general, an [latex]n[\/latex]th root is a number which, when raised to the [latex]n[\/latex]th power, gives the original number. You can only take an even root of numbers that are greater than or equal to\u00a0[latex]0[\/latex]. When simplifying radical expressions with variables, be careful to use absolute values when the index is even.<\/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-187\">\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><\/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":395986,"menu_order":1,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Simplify a Variety of Square Expressions (Simplify 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