{"id":16420,"date":"2019-10-03T15:21:09","date_gmt":"2019-10-03T15:21:09","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/7-1-2-squares-cubes-and-beyond\/"},"modified":"2024-05-02T15:38:45","modified_gmt":"2024-05-02T15:38:45","slug":"7-1-2-squares-cubes-and-beyond","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/7-1-2-squares-cubes-and-beyond\/","title":{"raw":"Cube Roots and Nth Roots","rendered":"Cube Roots and Nth Roots"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Define and simplify cube\u00a0roots<\/li>\r\n \t<li>Define and evaluate nth roots<\/li>\r\n \t<li>Estimate roots that are not perfect<\/li>\r\n<\/ul>\r\n<\/div>\r\n\r\n[caption id=\"attachment_5110\" align=\"aligncenter\" width=\"289\"]<img class=\"size-medium wp-image-5110\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/06\/23013616\/Screen-Shot-2016-06-22-at-6.35.40-PM-289x300.png\" alt=\"Rubik's cube\" width=\"289\" height=\"300\" \/> Rubik's Cune[\/caption]\r\n\r\nWe know that [latex]5^2=25, \\text{ and }\\sqrt{25}=5[\/latex], but what if we want to \"undo\" [latex]5^3=125, \\text{ or }5^4=625[\/latex]? We can use higher order roots to answer these questions.\r\n\r\nWhile square roots are probably the most common radical, you can also find the third root, the fifth root, the [latex]10th[\/latex]\u00a0root, or really any other <i>n<\/i>th root of a number. Just as the square root is a number that, when squared, gives the radicand, the <strong>cube root<\/strong> is a number that, when cubed, gives the radicand.\r\n\r\nThe cube root of a number is written with a small number \u00a0[latex]3[\/latex], called the <strong>index<\/strong>, just outside and above the radical symbol. It looks like [latex] \\sqrt[3]{{}}[\/latex]. This little [latex]3[\/latex] distinguishes cube roots from square roots which are written without a small number outside and above the radical symbol.\r\n<div class=\"textbox shaded\">\r\n\r\n<img class=\" wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-300x265.png\" alt=\"Caution\" width=\"50\" height=\"44\" \/>Caution! Be careful to distinguish between [latex] \\sqrt[3]{x}[\/latex], the cube root of <i>x<\/i>, and [latex] 3\\sqrt{x}[\/latex], three <i>times<\/i> the <i>square<\/i> root of <i>x<\/i>. They may look similar at first, but they lead you to much different expressions!\r\n\r\n<\/div>\r\nSuppose we know that [latex]{a}^{3}=8[\/latex]. We want to find what number raised to the\u00a0[latex]3[\/latex]rd power is equal to\u00a0[latex]8[\/latex]. Since [latex]{2}^{3}=8[\/latex], we say that\u00a0[latex]2[\/latex] is the cube root of\u00a0[latex]8[\/latex]. In the next example, we will evaluate the cube roots of some perfect cubes.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nEvaluate the following:\r\n<ol>\r\n \t<li>[latex] \\sqrt[3]{-8}[\/latex]<\/li>\r\n \t<li>[latex] \\sqrt[3]{27}[\/latex]<\/li>\r\n \t<li>[latex]0[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"517592\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"517592\"]\r\n\r\n1. We want to find a number whose cube is\u00a0[latex]-8[\/latex]. We know\u00a0[latex]2[\/latex] is the cube root of\u00a0[latex]8[\/latex], so maybe we can try\u00a0[latex]-2[\/latex] which gives [latex]-2\\cdot{-2}\\cdot{-2}=-8[\/latex], so the cube root of\u00a0[latex]-8[\/latex] is\u00a0[latex]-2[\/latex]. This is different from square roots because multiplying three negative numbers together results in a negative number.\r\n\r\n2. We want to find a number whose cube is\u00a0[latex]27[\/latex]. [latex]3\\cdot{3}\\cdot{3}=27[\/latex], so the cube root of [latex]27[\/latex] is [latex]3[\/latex].\r\n\r\n3.\u00a0 We want to find a number whose cube is [latex]0[\/latex]. [latex]0\\cdot0\\cdot0[\/latex], no matter how many times you multiply [latex]0[\/latex] by itself, you will always get\u00a0[latex]0[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nAs we saw in the last example, there is one interesting fact about cube roots that is not true of square roots. Negative numbers cannot have real number square roots, but negative numbers can have real number cube roots! What is the cube root of [latex]\u22128[\/latex]? [latex] \\sqrt[3]{-8}=-2[\/latex] because [latex] -2\\cdot -2\\cdot -2=-8[\/latex]. Remember, when you are multiplying an odd number of negative numbers, the result is negative! Consider [latex] \\sqrt[3]{{{(-1)}^{3}}}=-1[\/latex].\r\n\r\nWe can also use factoring to simplify cube roots such as [latex] \\sqrt[3]{125}[\/latex]. You can read this as \u201cthe third root of [latex]125[\/latex]\u201d or \u201cthe cube root of [latex]125[\/latex].\u201d To simplify this expression, look for a number that, when multiplied by itself two times (for a total of three identical factors), equals [latex]125[\/latex]. Let\u2019s factor [latex]125[\/latex] and find that number.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\sqrt[3]{125}[\/latex]\r\n\r\n[reveal-answer q=\"518592\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"518592\"][latex]125[\/latex] ends in [latex]5[\/latex], so you know that \u00a0[latex]5[\/latex] is a factor. Expand [latex]125[\/latex] into [latex]5\\cdot25[\/latex].\r\n<p style=\"text-align: center;\">[latex] \\sqrt[3]{5\\cdot 25}[\/latex]<\/p>\r\nFactor [latex]25[\/latex] into [latex]5[\/latex]and [latex]5[\/latex].\r\n<p style=\"text-align: center;\">[latex] \\sqrt[3]{5\\cdot 5\\cdot 5}[\/latex]<\/p>\r\nThe factors are [latex]5\\cdot5\\cdot5[\/latex], or [latex]5^{3}[\/latex].\r\n<p style=\"text-align: center;\">[latex] \\sqrt[3]{{{5}^{3}}}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex] \\sqrt[3]{125}=5[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe prime factors of [latex]125[\/latex] are [latex]5\\cdot5\\cdot5[\/latex], which can be rewritten as [latex]5^{3}[\/latex]. The cube root of a cubed number is the number itself, so [latex] \\sqrt[3]{{{5}^{3}}}=5[\/latex]. You have found the cube root, the three identical factors that when multiplied together give [latex]125[\/latex]. [latex]125[\/latex] is known as a <strong>perfect cube<\/strong> because its cube root is an integer.\r\n\r\nIn the following video, we show more examples of finding a cube root.\r\n\r\nhttps:\/\/youtu.be\/9Nh-Ggd2VJo\r\n\r\nHere\u2019s an example of how to simplify a radical that is not a perfect cube.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\sqrt[3]{32{{m}^{5}}}[\/latex]\r\n\r\n[reveal-answer q=\"617053\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"617053\"]Factor [latex]32[\/latex] into prime factors.\r\n<p style=\"text-align: center;\">[latex] \\sqrt[3]{2\\cdot 2\\cdot 2\\cdot 2\\cdot 2\\cdot {{m}^{5}}}[\/latex]<\/p>\r\nSince you are looking for the cube root, you need to find factors that appear [latex]3[\/latex] times under the radical. Rewrite [latex] 2\\cdot 2\\cdot 2[\/latex] as [latex] {{2}^{3}}[\/latex].\r\n<p style=\"text-align: center;\">[latex] \\sqrt[3]{{{2}^{3}}\\cdot 2\\cdot 2\\cdot {{m}^{5}}}[\/latex]<\/p>\r\nRewrite [latex] {{m}^{5}}[\/latex] as [latex] {{m}^{3}}\\cdot {{m}^{2}}[\/latex].\r\n<p style=\"text-align: center;\">[latex] \\sqrt[3]{{{2}^{3}}\\cdot 2\\cdot 2\\cdot {{m}^{3}}\\cdot {{m}^{2}}}[\/latex]<\/p>\r\nRewrite the expression as a product of multiple radicals.\r\n<p style=\"text-align: center;\">[latex] \\sqrt[3]{{{2}^{3}}}\\cdot \\sqrt[3]{2\\cdot 2}\\cdot \\sqrt[3]{{{m}^{3}}}\\cdot \\sqrt[3]{{{m}^{2}}}[\/latex]<\/p>\r\nSimplify and multiply.\r\n<p style=\"text-align: center;\">[latex] 2\\cdot \\sqrt[3]{4}\\cdot m\\cdot \\sqrt[3]{{{m}^{2}}}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex] \\sqrt[3]{32{{m}^{5}}}=2m\\sqrt[3]{4{{m}^{2}}}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the example below, we use the idea that\u00a0\u00a0[latex] \\sqrt[3]{{{(-1)}^{3}}}=-1[\/latex] to simplify the radical. \u00a0You do not have to do this, but it may help you recognize cubes more easily when they are nonnegative.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\sqrt[3]{-27{{x}^{4}}{{y}^{3}}}[\/latex]\r\n\r\n[reveal-answer q=\"670300\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"670300\"]Factor the expression into cubes.\r\n\r\nSeparate the cubed factors into individual radicals.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\sqrt[3]{-1\\cdot 27\\cdot {{x}^{4}}\\cdot {{y}^{3}}}\\\\\\sqrt[3]{{{(-1)}^{3}}\\cdot {{(3)}^{3}}\\cdot {{x}^{3}}\\cdot x\\cdot {{y}^{3}}}\\\\\\sqrt[3]{{{(-1)}^{3}}}\\cdot \\sqrt[3]{{{(3)}^{3}}}\\cdot \\sqrt[3]{{{x}^{3}}}\\cdot \\sqrt[3]{x}\\cdot \\sqrt[3]{{{y}^{3}}}\\end{array}[\/latex]<\/p>\r\nSimplify the cube roots.\r\n<p style=\"text-align: center;\">[latex] -1\\cdot 3\\cdot x\\cdot y\\cdot \\sqrt[3]{x}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex] \\sqrt[3]{-27{{x}^{4}}{{y}^{3}}}=-3xy\\sqrt[3]{x}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nYou could check your answer by performing the inverse operation. If you are right, when you cube [latex] -3xy\\sqrt[3]{x}[\/latex] you should get [latex] -27{{x}^{4}}{{y}^{3}}[\/latex].\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{l}\\left( -3xy\\sqrt[3]{x} \\right)\\left( -3xy\\sqrt[3]{x} \\right)\\left( -3xy\\sqrt[3]{x} \\right)\\\\-3\\cdot -3\\cdot -3\\cdot x\\cdot x\\cdot x\\cdot y\\cdot y\\cdot y\\cdot \\sqrt[3]{x}\\cdot \\sqrt[3]{x}\\cdot \\sqrt[3]{x}\\\\-27\\cdot {{x}^{3}}\\cdot {{y}^{3}}\\cdot \\sqrt[3]{{{x}^{3}}}\\\\-27{{x}^{3}}{{y}^{3}}\\cdot x\\\\-27{{x}^{4}}{{y}^{3}}\\end{array}[\/latex]<\/p>\r\nYou can find the odd root of a negative number, but you cannot find the even root of a negative number. This means you can simplify the radicals [latex] \\sqrt[3]{-81},\\ \\sqrt[5]{-64}[\/latex], and [latex] \\sqrt[7]{-2187}[\/latex], but you cannot simplify the radicals [latex] \\sqrt[{}]{-100},\\ \\sqrt[4]{-16}[\/latex], or [latex] \\sqrt[6]{-2,500}[\/latex].\r\n\r\nLet\u2019s look at another example.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\sqrt[3]{-24{{a}^{5}}}[\/latex]\r\n\r\n[reveal-answer q=\"473861\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"473861\"]Factor [latex]\u221224[\/latex] to find perfect cubes. Here, [latex]\u22121[\/latex] and 8 are the perfect cubes.\r\n<p style=\"text-align: center;\">[latex] \\sqrt[3]{-1\\cdot 8\\cdot 3\\cdot {{a}^{5}}}[\/latex]<\/p>\r\nFactor variables. You are looking\u00a0for cube exponents, so you factor\u00a0[latex]a^{5}[\/latex]\u00a0into [latex]a^{3}[\/latex]\u00a0and [latex]a^{2}[\/latex].\r\n<p style=\"text-align: center;\">[latex] \\sqrt[3]{{{(-1)}^{3}}\\cdot {{2}^{3}}\\cdot 3\\cdot {{a}^{3}}\\cdot {{a}^{2}}}[\/latex]<\/p>\r\nSeparate the factors into individual radicals.\r\n<p style=\"text-align: center;\">[latex] \\sqrt[3]{{{(-1)}^{3}}}\\cdot \\sqrt[3]{{{2}^{3}}}\\cdot \\sqrt[3]{{{a}^{3}}}\\cdot \\sqrt[3]{3\\cdot {{a}^{2}}}[\/latex]<\/p>\r\nSimplify, using the property [latex] \\sqrt[3]{{{x}^{3}}}=x[\/latex].<em>\u00a0<\/em>\r\n<p style=\"text-align: center;\">[latex] -1\\cdot 2\\cdot a\\cdot \\sqrt[3]{3\\cdot {{a}^{2}}}[\/latex]<\/p>\r\nThis is the simplest form of this expression; all cubes have been pulled out of the radical expression.\r\n<p style=\"text-align: center;\">[latex] -2a\\sqrt[3]{3{{a}^{2}}}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex] \\sqrt[3]{-24{{a}^{5}}}=-2a\\sqrt[3]{3{{a}^{2}}}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the next video, we share examples of finding cube roots with negative radicands.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]196049[\/ohm_question]\r\n\r\n<\/div>\r\nhttps:\/\/youtu.be\/BtJruOpmHCE\r\n\r\nIn the same way that we learned earlier that we can estimate square roots, we can also estimate cube roots.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nApproximate [latex] \\sqrt[3]{30}[\/latex] and also find its value using a calculator.\r\n\r\n[reveal-answer q=\"71092\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"71092\"]\r\n\r\nFind the cubes that surround\u00a0[latex]30[\/latex].\r\n\r\n[latex]30[\/latex] is in between the perfect cubes\u00a0[latex]27[\/latex] and\u00a0[latex]81[\/latex].\r\n\r\n[latex] \\sqrt[3]{27}=3[\/latex] and [latex] \\sqrt[3]{81}=4[\/latex], so [latex] \\sqrt[3]{30}[\/latex] is between\u00a0[latex]3[\/latex] and\u00a0[latex]4[\/latex].\r\nUse a calculator.\r\n<p style=\"text-align: center;\">[latex]\\sqrt[3]{30}\\approx3.10723[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]189478[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Nth Roots<\/h2>\r\nWe learned above that the cube root of a number is written with a small number\u00a0[latex]3[\/latex], which looks like [latex] \\sqrt[3]{a}[\/latex]. This number placed just outside and above the radical symbol and is called the <strong>index.<\/strong>\u00a0This little\u00a0[latex]3[\/latex] distinguishes cube roots from square roots which are written without a small number outside and above the radical symbol.\r\n\r\nWe can apply the same idea to any exponent and its corresponding root. The <em>n<\/em>th root of [latex]a[\/latex] is a number that, when raised to the <em>n<\/em>th power, gives [latex]a[\/latex]. For example, [latex]3[\/latex] is the 5th root of [latex]243[\/latex] because [latex]{\\left(3\\right)}^{5}=243[\/latex]. If [latex]a[\/latex] is a real number with at least one <em>n<\/em>th root, then the <strong>principal <em>n<\/em>th root<\/strong> of [latex]a[\/latex] is the number with the same sign as [latex]a[\/latex] that, when raised to the <em>n<\/em>th power, equals [latex]a[\/latex].\r\n\r\nThe principal <em>n<\/em>th root of [latex]a[\/latex] is written as [latex]\\sqrt[n]{a}[\/latex], where [latex]n[\/latex] is a positive integer greater than or equal to\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\">\r\n<h3>Definition:\u00a0Principal <em>n<\/em>th Root<\/h3>\r\nIf [latex]a[\/latex] is a real number with at least one <em>n<\/em>th root, then the <strong>principal <em>n<\/em>th root<\/strong> of [latex]a[\/latex], written as [latex]\\sqrt[n]{a}[\/latex], is the number with the same sign as [latex]a[\/latex] that, when raised to the <em>n<\/em>th power, equals [latex]a[\/latex]. The <strong>index<\/strong> of the radical is [latex]n[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nEvaluate each of the following:\r\n<ol>\r\n \t<li>[latex]\\sqrt[5]{-32}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt[4]{81}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt[8]{-1}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"140298\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"140298\"]\r\n<ol>\r\n \t<li>[latex]\\sqrt[5]{-32}[\/latex] Factor\u00a0[latex]32[\/latex], which gives [latex]{\\left(-2\\right)}^{5}=-32 \\\\ \\text{ }[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt[4]{81}[\/latex]. Factoring can help. We know that [latex]9\\cdot9=81[\/latex] and we can further factor each\u00a0[latex]9[\/latex]: [latex]\\sqrt[4]{81}=\\sqrt[4]{3\\cdot3\\cdot3\\cdot3}=\\sqrt[4]{3^4}=3[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt[8]{-1}[\/latex]. Since we have an\u00a0[latex]8[\/latex]th root - which is even- with a negative number as the radicand, this root has no real number solutions. In other words, [latex]-1\\cdot-1\\cdot-1\\cdot-1\\cdot-1\\cdot-1\\cdot-1\\cdot-1=+1[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe steps to consider when simplifying a radical are outlined below.\r\n<div class=\"textbox shaded\">\r\n<h3>Simplifying a radical<\/h3>\r\nWhen working with exponents and radicals:\r\n<ul>\r\n \t<li>If <i>n<\/i> is odd, [latex] \\sqrt[n]{{{x}^{n}}}=x[\/latex].<\/li>\r\n \t<li>If <i>n<\/i> is even, [latex] \\sqrt[n]{{{x}^{n}}}=\\left| x \\right|[\/latex]. (The absolute value accounts for the fact that if <i>x<\/i> is negative and raised to an even power, that number will be positive, as will the <i>n<\/i>th principal root of that number.)<\/li>\r\n<\/ul>\r\n<\/div>\r\nIn the following video, we show more examples of how to evaluate <em>n<\/em>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]{-81},\\ \\sqrt[5]{-64}[\/latex], and [latex] \\sqrt[7]{-2187}[\/latex] because the all have an odd numbered index, but you cannot evaluate the radicals [latex] \\sqrt[{}]{-100},\\ \\sqrt[4]{-16}[\/latex], or [latex] \\sqrt[6]{-2,500}[\/latex] because they all have an even numbered index.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]94221[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Summary<\/h2>\r\nA radical expression is a mathematical way of representing the <i>n<\/i>th root of a number. Square roots and cube roots are the most common radicals, but a root can be any number. To simplify radical expressions, look for exponential factors within the radical, and then use the property [latex] \\sqrt[n]{{{x}^{n}}}=x[\/latex] if <i>n<\/i> is odd, and [latex] \\sqrt[n]{{{x}^{n}}}=\\left| x \\right|[\/latex] if <i>n<\/i> is even to pull out quantities. All rules of integer operations and exponents apply when simplifying radical expressions.\u00a0\u00a0Nth roots can be approximated using trial and error or a calculator.","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Define and simplify cube\u00a0roots<\/li>\n<li>Define and evaluate nth roots<\/li>\n<li>Estimate roots that are not perfect<\/li>\n<\/ul>\n<\/div>\n<div id=\"attachment_5110\" style=\"width: 299px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-5110\" class=\"size-medium wp-image-5110\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/06\/23013616\/Screen-Shot-2016-06-22-at-6.35.40-PM-289x300.png\" alt=\"Rubik's cube\" width=\"289\" height=\"300\" \/><\/p>\n<p id=\"caption-attachment-5110\" class=\"wp-caption-text\">Rubik&#8217;s Cune<\/p>\n<\/div>\n<p>We know that [latex]5^2=25, \\text{ and }\\sqrt{25}=5[\/latex], but what if we want to &#8220;undo&#8221; [latex]5^3=125, \\text{ or }5^4=625[\/latex]? We can use higher order roots to answer these questions.<\/p>\n<p>While square roots are probably the most common radical, you can also find the third root, the fifth root, the [latex]10th[\/latex]\u00a0root, or really any other <i>n<\/i>th root of a number. Just as the square root is a number that, when squared, gives the radicand, the <strong>cube root<\/strong> is a number that, when cubed, gives the radicand.<\/p>\n<p>The cube root of a number is written with a small number \u00a0[latex]3[\/latex], called the <strong>index<\/strong>, just outside and above the radical symbol. It looks like [latex]\\sqrt[3]{{}}[\/latex]. This little [latex]3[\/latex] distinguishes cube roots from square roots which are written without a small number outside and above the radical symbol.<\/p>\n<div class=\"textbox shaded\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-300x265.png\" alt=\"Caution\" width=\"50\" height=\"44\" \/>Caution! Be careful to distinguish between [latex]\\sqrt[3]{x}[\/latex], the cube root of <i>x<\/i>, and [latex]3\\sqrt{x}[\/latex], three <i>times<\/i> the <i>square<\/i> root of <i>x<\/i>. They may look similar at first, but they lead you to much different expressions!<\/p>\n<\/div>\n<p>Suppose we know that [latex]{a}^{3}=8[\/latex]. We want to find what number raised to the\u00a0[latex]3[\/latex]rd power is equal to\u00a0[latex]8[\/latex]. Since [latex]{2}^{3}=8[\/latex], we say that\u00a0[latex]2[\/latex] is the cube root of\u00a0[latex]8[\/latex]. In the next example, we will evaluate the cube roots of some perfect cubes.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Evaluate the following:<\/p>\n<ol>\n<li>[latex]\\sqrt[3]{-8}[\/latex]<\/li>\n<li>[latex]\\sqrt[3]{27}[\/latex]<\/li>\n<li>[latex]0[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q517592\">Show Solution<\/span><\/p>\n<div id=\"q517592\" class=\"hidden-answer\" style=\"display: none\">\n<p>1. We want to find a number whose cube is\u00a0[latex]-8[\/latex]. We know\u00a0[latex]2[\/latex] is the cube root of\u00a0[latex]8[\/latex], so maybe we can try\u00a0[latex]-2[\/latex] which gives [latex]-2\\cdot{-2}\\cdot{-2}=-8[\/latex], so the cube root of\u00a0[latex]-8[\/latex] is\u00a0[latex]-2[\/latex]. This is different from square roots because multiplying three negative numbers together results in a negative number.<\/p>\n<p>2. We want to find a number whose cube is\u00a0[latex]27[\/latex]. [latex]3\\cdot{3}\\cdot{3}=27[\/latex], so the cube root of [latex]27[\/latex] is [latex]3[\/latex].<\/p>\n<p>3.\u00a0 We want to find a number whose cube is [latex]0[\/latex]. [latex]0\\cdot0\\cdot0[\/latex], no matter how many times you multiply [latex]0[\/latex] by itself, you will always get\u00a0[latex]0[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>As we saw in the last example, there is one interesting fact about cube roots that is not true of square roots. Negative numbers cannot have real number square roots, but negative numbers can have real number cube roots! What is the cube root of [latex]\u22128[\/latex]? [latex]\\sqrt[3]{-8}=-2[\/latex] because [latex]-2\\cdot -2\\cdot -2=-8[\/latex]. Remember, when you are multiplying an odd number of negative numbers, the result is negative! Consider [latex]\\sqrt[3]{{{(-1)}^{3}}}=-1[\/latex].<\/p>\n<p>We can also use factoring to simplify cube roots such as [latex]\\sqrt[3]{125}[\/latex]. You can read this as \u201cthe third root of [latex]125[\/latex]\u201d or \u201cthe cube root of [latex]125[\/latex].\u201d To simplify this expression, look for a number that, when multiplied by itself two times (for a total of three identical factors), equals [latex]125[\/latex]. Let\u2019s factor [latex]125[\/latex] and find that number.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\sqrt[3]{125}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q518592\">Show Solution<\/span><\/p>\n<div id=\"q518592\" class=\"hidden-answer\" style=\"display: none\">[latex]125[\/latex] ends in [latex]5[\/latex], so you know that \u00a0[latex]5[\/latex] is a factor. Expand [latex]125[\/latex] into [latex]5\\cdot25[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt[3]{5\\cdot 25}[\/latex]<\/p>\n<p>Factor [latex]25[\/latex] into [latex]5[\/latex]and [latex]5[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt[3]{5\\cdot 5\\cdot 5}[\/latex]<\/p>\n<p>The factors are [latex]5\\cdot5\\cdot5[\/latex], or [latex]5^{3}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt[3]{{{5}^{3}}}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\sqrt[3]{125}=5[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The prime factors of [latex]125[\/latex] are [latex]5\\cdot5\\cdot5[\/latex], which can be rewritten as [latex]5^{3}[\/latex]. The cube root of a cubed number is the number itself, so [latex]\\sqrt[3]{{{5}^{3}}}=5[\/latex]. You have found the cube root, the three identical factors that when multiplied together give [latex]125[\/latex]. [latex]125[\/latex] is known as a <strong>perfect cube<\/strong> because its cube root is an integer.<\/p>\n<p>In the following video, we show more examples of finding a cube root.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Simplify Cube Roots (Perfect Cube Radicands)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/9Nh-Ggd2VJo?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>Here\u2019s an example of how to simplify a radical that is not a perfect cube.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\sqrt[3]{32{{m}^{5}}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q617053\">Show Solution<\/span><\/p>\n<div id=\"q617053\" class=\"hidden-answer\" style=\"display: none\">Factor [latex]32[\/latex] into prime factors.<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt[3]{2\\cdot 2\\cdot 2\\cdot 2\\cdot 2\\cdot {{m}^{5}}}[\/latex]<\/p>\n<p>Since you are looking for the cube root, you need to find factors that appear [latex]3[\/latex] times under the radical. Rewrite [latex]2\\cdot 2\\cdot 2[\/latex] as [latex]{{2}^{3}}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt[3]{{{2}^{3}}\\cdot 2\\cdot 2\\cdot {{m}^{5}}}[\/latex]<\/p>\n<p>Rewrite [latex]{{m}^{5}}[\/latex] as [latex]{{m}^{3}}\\cdot {{m}^{2}}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt[3]{{{2}^{3}}\\cdot 2\\cdot 2\\cdot {{m}^{3}}\\cdot {{m}^{2}}}[\/latex]<\/p>\n<p>Rewrite the expression as a product of multiple radicals.<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt[3]{{{2}^{3}}}\\cdot \\sqrt[3]{2\\cdot 2}\\cdot \\sqrt[3]{{{m}^{3}}}\\cdot \\sqrt[3]{{{m}^{2}}}[\/latex]<\/p>\n<p>Simplify and multiply.<\/p>\n<p style=\"text-align: center;\">[latex]2\\cdot \\sqrt[3]{4}\\cdot m\\cdot \\sqrt[3]{{{m}^{2}}}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\sqrt[3]{32{{m}^{5}}}=2m\\sqrt[3]{4{{m}^{2}}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the example below, we use the idea that\u00a0\u00a0[latex]\\sqrt[3]{{{(-1)}^{3}}}=-1[\/latex] to simplify the radical. \u00a0You do not have to do this, but it may help you recognize cubes more easily when they are nonnegative.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\sqrt[3]{-27{{x}^{4}}{{y}^{3}}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q670300\">Show Solution<\/span><\/p>\n<div id=\"q670300\" class=\"hidden-answer\" style=\"display: none\">Factor the expression into cubes.<\/p>\n<p>Separate the cubed factors into individual radicals.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\sqrt[3]{-1\\cdot 27\\cdot {{x}^{4}}\\cdot {{y}^{3}}}\\\\\\sqrt[3]{{{(-1)}^{3}}\\cdot {{(3)}^{3}}\\cdot {{x}^{3}}\\cdot x\\cdot {{y}^{3}}}\\\\\\sqrt[3]{{{(-1)}^{3}}}\\cdot \\sqrt[3]{{{(3)}^{3}}}\\cdot \\sqrt[3]{{{x}^{3}}}\\cdot \\sqrt[3]{x}\\cdot \\sqrt[3]{{{y}^{3}}}\\end{array}[\/latex]<\/p>\n<p>Simplify the cube roots.<\/p>\n<p style=\"text-align: center;\">[latex]-1\\cdot 3\\cdot x\\cdot y\\cdot \\sqrt[3]{x}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\sqrt[3]{-27{{x}^{4}}{{y}^{3}}}=-3xy\\sqrt[3]{x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>You could check your answer by performing the inverse operation. If you are right, when you cube [latex]-3xy\\sqrt[3]{x}[\/latex] you should get [latex]-27{{x}^{4}}{{y}^{3}}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\left( -3xy\\sqrt[3]{x} \\right)\\left( -3xy\\sqrt[3]{x} \\right)\\left( -3xy\\sqrt[3]{x} \\right)\\\\-3\\cdot -3\\cdot -3\\cdot x\\cdot x\\cdot x\\cdot y\\cdot y\\cdot y\\cdot \\sqrt[3]{x}\\cdot \\sqrt[3]{x}\\cdot \\sqrt[3]{x}\\\\-27\\cdot {{x}^{3}}\\cdot {{y}^{3}}\\cdot \\sqrt[3]{{{x}^{3}}}\\\\-27{{x}^{3}}{{y}^{3}}\\cdot x\\\\-27{{x}^{4}}{{y}^{3}}\\end{array}[\/latex]<\/p>\n<p>You can find the odd root of a negative number, but you cannot find the even root of a negative number. This means you can simplify the radicals [latex]\\sqrt[3]{-81},\\ \\sqrt[5]{-64}[\/latex], and [latex]\\sqrt[7]{-2187}[\/latex], but you cannot simplify the radicals [latex]\\sqrt[{}]{-100},\\ \\sqrt[4]{-16}[\/latex], or [latex]\\sqrt[6]{-2,500}[\/latex].<\/p>\n<p>Let\u2019s look at another example.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\sqrt[3]{-24{{a}^{5}}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q473861\">Show Solution<\/span><\/p>\n<div id=\"q473861\" class=\"hidden-answer\" style=\"display: none\">Factor [latex]\u221224[\/latex] to find perfect cubes. Here, [latex]\u22121[\/latex] and 8 are the perfect cubes.<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt[3]{-1\\cdot 8\\cdot 3\\cdot {{a}^{5}}}[\/latex]<\/p>\n<p>Factor variables. You are looking\u00a0for cube exponents, so you factor\u00a0[latex]a^{5}[\/latex]\u00a0into [latex]a^{3}[\/latex]\u00a0and [latex]a^{2}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt[3]{{{(-1)}^{3}}\\cdot {{2}^{3}}\\cdot 3\\cdot {{a}^{3}}\\cdot {{a}^{2}}}[\/latex]<\/p>\n<p>Separate the factors into individual radicals.<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt[3]{{{(-1)}^{3}}}\\cdot \\sqrt[3]{{{2}^{3}}}\\cdot \\sqrt[3]{{{a}^{3}}}\\cdot \\sqrt[3]{3\\cdot {{a}^{2}}}[\/latex]<\/p>\n<p>Simplify, using the property [latex]\\sqrt[3]{{{x}^{3}}}=x[\/latex].<em>\u00a0<\/em><\/p>\n<p style=\"text-align: center;\">[latex]-1\\cdot 2\\cdot a\\cdot \\sqrt[3]{3\\cdot {{a}^{2}}}[\/latex]<\/p>\n<p>This is the simplest form of this expression; all cubes have been pulled out of the radical expression.<\/p>\n<p style=\"text-align: center;\">[latex]-2a\\sqrt[3]{3{{a}^{2}}}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\sqrt[3]{-24{{a}^{5}}}=-2a\\sqrt[3]{3{{a}^{2}}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the next video, we share examples of finding cube roots with negative radicands.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm196049\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=196049&theme=oea&iframe_resize_id=ohm196049&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Simplify Cube Roots (Not Perfect Cube Radicands)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/BtJruOpmHCE?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>In the same way that we learned earlier that we can estimate square roots, we can also estimate cube roots.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Approximate [latex]\\sqrt[3]{30}[\/latex] and also find its value using a calculator.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q71092\">Show Solution<\/span><\/p>\n<div id=\"q71092\" class=\"hidden-answer\" style=\"display: none\">\n<p>Find the cubes that surround\u00a0[latex]30[\/latex].<\/p>\n<p>[latex]30[\/latex] is in between the perfect cubes\u00a0[latex]27[\/latex] and\u00a0[latex]81[\/latex].<\/p>\n<p>[latex]\\sqrt[3]{27}=3[\/latex] and [latex]\\sqrt[3]{81}=4[\/latex], so [latex]\\sqrt[3]{30}[\/latex] is between\u00a0[latex]3[\/latex] and\u00a0[latex]4[\/latex].<br \/>\nUse a calculator.<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt[3]{30}\\approx3.10723[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm189478\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=189478&theme=oea&iframe_resize_id=ohm189478&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Nth Roots<\/h2>\n<p>We learned above that the cube root of a number is written with a small number\u00a0[latex]3[\/latex], which looks like [latex]\\sqrt[3]{a}[\/latex]. This number placed just outside and above the radical symbol and is called the <strong>index.<\/strong>\u00a0This little\u00a0[latex]3[\/latex] distinguishes cube roots from square roots which are written without a small number outside and above the radical symbol.<\/p>\n<p>We can apply the same idea to any exponent and its corresponding root. The <em>n<\/em>th root of [latex]a[\/latex] is a number that, when raised to the <em>n<\/em>th power, gives [latex]a[\/latex]. For example, [latex]3[\/latex] is the 5th root of [latex]243[\/latex] because [latex]{\\left(3\\right)}^{5}=243[\/latex]. If [latex]a[\/latex] is a real number with at least one <em>n<\/em>th root, then the <strong>principal <em>n<\/em>th root<\/strong> of [latex]a[\/latex] is the number with the same sign as [latex]a[\/latex] that, when raised to the <em>n<\/em>th power, equals [latex]a[\/latex].<\/p>\n<p>The principal <em>n<\/em>th root of [latex]a[\/latex] is written as [latex]\\sqrt[n]{a}[\/latex], where [latex]n[\/latex] is a positive integer greater than or equal to\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\">\n<h3>Definition:\u00a0Principal <em>n<\/em>th Root<\/h3>\n<p>If [latex]a[\/latex] is a real number with at least one <em>n<\/em>th root, then the <strong>principal <em>n<\/em>th root<\/strong> of [latex]a[\/latex], written as [latex]\\sqrt[n]{a}[\/latex], is the number with the same sign as [latex]a[\/latex] that, when raised to the <em>n<\/em>th power, equals [latex]a[\/latex]. The <strong>index<\/strong> of the radical is [latex]n[\/latex].<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Evaluate each of the following:<\/p>\n<ol>\n<li>[latex]\\sqrt[5]{-32}[\/latex]<\/li>\n<li>[latex]\\sqrt[4]{81}[\/latex]<\/li>\n<li>[latex]\\sqrt[8]{-1}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q140298\">Show Solution<\/span><\/p>\n<div id=\"q140298\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]\\sqrt[5]{-32}[\/latex] Factor\u00a0[latex]32[\/latex], which gives [latex]{\\left(-2\\right)}^{5}=-32 \\\\ \\text{ }[\/latex]<\/li>\n<li>[latex]\\sqrt[4]{81}[\/latex]. Factoring can help. We know that [latex]9\\cdot9=81[\/latex] and we can further factor each\u00a0[latex]9[\/latex]: [latex]\\sqrt[4]{81}=\\sqrt[4]{3\\cdot3\\cdot3\\cdot3}=\\sqrt[4]{3^4}=3[\/latex]<\/li>\n<li>[latex]\\sqrt[8]{-1}[\/latex]. Since we have an\u00a0[latex]8[\/latex]th root &#8211; which is even- with a negative number as the radicand, this root has no real number solutions. In other words, [latex]-1\\cdot-1\\cdot-1\\cdot-1\\cdot-1\\cdot-1\\cdot-1\\cdot-1=+1[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>The steps to consider when simplifying a radical are outlined below.<\/p>\n<div class=\"textbox shaded\">\n<h3>Simplifying a radical<\/h3>\n<p>When working with exponents and radicals:<\/p>\n<ul>\n<li>If <i>n<\/i> is odd, [latex]\\sqrt[n]{{{x}^{n}}}=x[\/latex].<\/li>\n<li>If <i>n<\/i> is even, [latex]\\sqrt[n]{{{x}^{n}}}=\\left| x \\right|[\/latex]. (The absolute value accounts for the fact that if <i>x<\/i> is negative and raised to an even power, that number will be positive, as will the <i>n<\/i>th principal root of that number.)<\/li>\n<\/ul>\n<\/div>\n<p>In the following video, we show more examples of how to evaluate <em>n<\/em>th roots.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Simplify Perfect Nth Roots\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/vA2DkcUSRSk?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can find the odd root of a negative number, but you cannot find the even root of a negative number. This means you can evaluate the radicals [latex]\\sqrt[3]{-81},\\ \\sqrt[5]{-64}[\/latex], and [latex]\\sqrt[7]{-2187}[\/latex] because the all have an odd numbered index, but you cannot evaluate the radicals [latex]\\sqrt[{}]{-100},\\ \\sqrt[4]{-16}[\/latex], or [latex]\\sqrt[6]{-2,500}[\/latex] because they all have an even numbered index.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm94221\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=94221&theme=oea&iframe_resize_id=ohm94221&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Summary<\/h2>\n<p>A radical expression is a mathematical way of representing the <i>n<\/i>th root of a number. Square roots and cube roots are the most common radicals, but a root can be any number. To simplify radical expressions, look for exponential factors within the radical, and then use the property [latex]\\sqrt[n]{{{x}^{n}}}=x[\/latex] if <i>n<\/i> is odd, and [latex]\\sqrt[n]{{{x}^{n}}}=\\left| x \\right|[\/latex] if <i>n<\/i> is even to pull out quantities. All rules of integer operations and exponents apply when simplifying radical expressions.\u00a0\u00a0Nth roots can be approximated using trial and error or a calculator.<\/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-16420\">\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>Screenshot: Rubik&#039;s Cube. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Svreenshot: Caution. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Simplify Cube Roots (Perfect Cube Radicands). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/9Nh-Ggd2VJo\">https:\/\/youtu.be\/9Nh-Ggd2VJo<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Simplify Cube Roots (Not Perfect Cube Radicands). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/BtJruOpmHCE\">https:\/\/youtu.be\/BtJruOpmHCE<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Unit 16: Radical Expressions and Quadratic Equations, from Developmental Math: An Open Program. <strong>Provided by<\/strong>: Monterey Institute of Technology and Education. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/\">http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":169554,"menu_order":5,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Screenshot: Rubik\\'s Cube\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Svreenshot: Caution\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"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 and Education\",\"url\":\" http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Simplify Cube Roots (Perfect Cube Radicands)\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/9Nh-Ggd2VJo\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Simplify Cube Roots (Not Perfect Cube Radicands)\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/BtJruOpmHCE\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"06f4479c8f51472f98284003fdedb3ce, e0327ea64b834051bc054a3d5638f560, 67ff3c2903ef41fb93315afeda4a4cca, fa6cdbc1066548d0ab5ae61d1ef6e4ae","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-16420","chapter","type-chapter","status-publish","hentry"],"part":16199,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/16420","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/users\/169554"}],"version-history":[{"count":10,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/16420\/revisions"}],"predecessor-version":[{"id":19962,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/16420\/revisions\/19962"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/parts\/16199"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/16420\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/media?parent=16420"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapter-type?post=16420"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/contributor?post=16420"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/license?post=16420"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}