{"id":5030,"date":"2017-12-31T00:09:13","date_gmt":"2017-12-31T00:09:13","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/cuny-hunter-collegealgebra\/?post_type=chapter&#038;p=5030"},"modified":"2023-11-08T13:20:19","modified_gmt":"2023-11-08T13:20:19","slug":"introduction-to-roots","status":"web-only","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/cuny-hunter-collegealgebra\/chapter\/introduction-to-roots\/","title":{"raw":"9.1- Introduction to Roots","rendered":"9.1- Introduction to Roots"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>(9.1.1) - Define and evaluate principal square roots\r\n<ul>\r\n \t<li>Square roots<\/li>\r\n \t<li>Cube roots<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>(9.1.2) - Define and evaluate n<sup>th<\/sup> roots<\/li>\r\n \t<li>(9.1.3) - Estimate roots that are not perfect<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h1>(9.1.1) - Define and evaluate principal square roots<\/h1>\r\nDid you know that you can take the 6th root of a number? You have probably heard of a square root, written [latex]\\sqrt{}[\/latex], but you can also take a third, fourth and even a 5,000th root (if you really had to). In this lesson we will learn how a square root is defined and then we will build on that to form an understanding of nth roots. \u00a0We will use factoring and rules for exponents to simplify mathematical expressions that contain roots.\r\n\r\nThe most common root is the <strong>square root<\/strong>. First, we will define what square roots are,\u00a0 and how you find the square root of a number. Then we will apply similar ideas to define and evaluate nth roots.\r\n\r\nRoots are the inverse of exponents, much like multiplication is the inverse of division. Recall\u00a0how exponents are defined, and written; with an exponent, as words, and as repeated multiplication.\r\n\r\n<strong>Exponent:<\/strong> [latex] {{3}^{2}}[\/latex],\u00a0[latex] {{4}^{5}}[\/latex],\u00a0[latex] {{x}^{3}}[\/latex],\u00a0[latex] {{x}^{\\text{n}}}[\/latex]\r\n\r\n<strong>Name:<\/strong>\u00a0\u201cThree squared\u201d or\u00a0\u201cThree to the second power\u201d,\u00a0\u201cFour to the fifth power\u201d,\u00a0\u201c<i>x<\/i> cubed\u201d,\u00a0\u201c<i>x<\/i> to the <i>n<\/i>th power\u201d\r\n\r\n<strong>Repeated Multiplication:<\/strong>\u00a0[latex] 3\\cdot 3[\/latex], \u00a0[latex] 4\\cdot 4\\cdot 4\\cdot 4\\cdot 4[\/latex], \u00a0[latex] x\\cdot x\\cdot x[\/latex], \u00a0[latex] \\underbrace{x\\cdot x\\cdot x...\\cdot x}_{n\\text{ times}}[\/latex].\r\n\r\nConversely,\u00a0 when you are trying to find the square root of a number (say, 25), you are trying to find a number that can be multiplied by itself to create that original number. In the case of 25, you can find that [latex]5\\cdot5=25[\/latex], so 5 must be the square root.\r\n<h3>Square Roots<\/h3>\r\nThe symbol for the square root is called a <strong>radical symbol<\/strong> and looks like this: [latex]\\sqrt{\\,\\,\\,}[\/latex]. The expression [latex] \\sqrt{25}[\/latex] is read \u201cthe square root of twenty-five\u201d or \u201cradical twenty-five.\u201d The number that is written under the radical symbol is called the <strong>radicand<\/strong>.\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200220\/CNX_CAT_Figure_01_03_002.jpg\" alt=\"The expression: square root of twenty-five is enclosed in a circle. The circle has an arrow pointing to it labeled: Radical expression. The square root symbol has an arrow pointing to it labeled: Radical. The number twenty-five has an arrow pointing to it labeled: Radicand.\" \/>\r\n\r\nThe following table shows different radicals and their equivalent written and simplified forms.\r\n<table style=\"width: 70%\">\r\n<thead>\r\n<tr>\r\n<th>Radical<\/th>\r\n<th>Name<\/th>\r\n<th>Simplified Form<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex] \\sqrt{36}[\/latex]<\/td>\r\n<td>\u201cSquare root of thirty-six\u201d\r\n\r\n\u201cRadical thirty-six\u201d<\/td>\r\n<td>[latex] \\sqrt{36}=\\sqrt{6\\cdot 6}=6[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex] \\sqrt{100}[\/latex]<\/td>\r\n<td>\u201cSquare root of one hundred\u201d\r\n\r\n\u201cRadical one hundred\u201d<\/td>\r\n<td>[latex] \\sqrt{100}=\\sqrt{10\\cdot 10}=10[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex] \\sqrt{225}[\/latex]<\/td>\r\n<td>\u201cSquare root of two hundred twenty-five\u201d\r\n\r\n\u201cRadical two hundred twenty-five\u201d<\/td>\r\n<td>[latex] \\sqrt{225}=\\sqrt{15\\cdot 15}=15[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nConsider [latex] \\sqrt{25}[\/latex] again. You may realize that there is another value that, when multiplied by itself, also results in 25. That number is [latex]\u22125[\/latex].\r\n<p style=\"text-align: center\">[latex] \\begin{array}{r}5\\cdot 5=25\\\\-5\\cdot -5=25\\end{array}[\/latex]<\/p>\r\nBy definition, the square root symbol always means to find the positive root, called the <strong>principal root<\/strong>. So while [latex]5\\cdot5[\/latex] and [latex]\u22125\\cdot\u22125[\/latex] both equal 25, only 5 is the principal root. You should also know that zero is special because it has only one square root: itself (since [latex]0\\cdot0=0[\/latex]).\r\n\r\nIn our first example we will show you how to use radical notation to evaluate principal square roots.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nFind the principal root of each expression.\r\n<ol>\r\n \t<li>[latex]\\sqrt{100}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{16}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{25+144}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{49}-\\sqrt{81}[\/latex]<\/li>\r\n \t<li>[latex] -\\sqrt{81}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{-9}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"419579\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"419579\"]\r\n<ol>\r\n \t<li>[latex]\\sqrt{100}=10[\/latex] because [latex]{10}^{2}=100[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{\\sqrt{16}}=\\sqrt{4}=2[\/latex] because [latex]{4}^{2}=16[\/latex] and [latex]{2}^{2}=4[\/latex]<\/li>\r\n \t<li>Recall that square roots act as grouping symbols in the order of operations, so addition and subtraction must be performed first when they occur under a radical. [latex]\\sqrt{25+144}=\\sqrt{169}=13[\/latex] because [latex]{13}^{2}=169[\/latex]<\/li>\r\n \t<li>This problem is similar to the last one, but this time subtraction should occur after evaluating the root. Stop and think about why these two problems are different. [latex]\\sqrt{49}-\\sqrt{81}=7 - 9=-2[\/latex] because [latex]{7}^{2}=49[\/latex] and [latex]{9}^{2}=81[\/latex]<\/li>\r\n \t<li>\r\n<p style=\"text-align: left\">The negative in front means to take the opposite of the value after you simplify the radical. [latex] -\\sqrt{81} = -\\sqrt{9\\cdot 9}[\/latex].\u00a0 The square root of [latex]81[\/latex] is [latex]9[\/latex]. Then, take the opposite of [latex]9:\u00a0 [latex]\u2212(9)[\/latex]<\/p>\r\n<\/li>\r\n \t<li>[latex]\\sqrt{-9}[\/latex], we are looking for a number that when it is squared, returns [latex]-9[\/latex]. We can try [latex](-3)^2[\/latex], but that will give a positive result, and [latex]3^2[\/latex] will also give a positive result. This leads to an important fact - \u00a0you cannot find the square root of a negative number.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video we present more examples of how to find a principle square root.\r\n\r\nhttps:\/\/youtu.be\/2cWAkmJoaDQ\r\n\r\nThe last example we showed leads to an important characteristic of square roots. You can only take the square root of values that are nonnegative.\r\n<p class=\"textbox shaded\"><strong>Domain of a Square Root<\/strong>\r\n[latex]\\sqrt{-a}[\/latex] is not defined for all real numbers, a. Therefore, [latex]\\sqrt{a}[\/latex] is defined for [latex]a\\ge0[\/latex]<\/p>\r\n\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Think About It<\/h3>\r\nDoes [latex]\\sqrt{25}=\\pm 5[\/latex]? Write your ideas and a sentence to defend them in the box below before you look at the answer.\r\n\r\n[practice-area rows=\"1\"][\/practice-area]\r\n[reveal-answer q=\"101071\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"101071\"]\r\n\r\n<em>No. Although both<\/em> [latex]{5}^{2}[\/latex] <em>and<\/em> [latex]{\\left(-5\\right)}^{2}[\/latex] <em>are<\/em> [latex]25[\/latex], <em>the radical symbol implies only a nonnegative root, the principal square root. The principal square root of 25 is<\/em> [latex]\\sqrt{25}=5[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3>Cube Roots<\/h3>\r\nWe know that [latex]5^2=25, \\text{ and }\\sqrt{25}=5[\/latex] but what if we want to \"undo\" [latex]5^3=125, \\text{ or }5^4=625[\/latex]? We can use higher order roots to answer these questions.\r\n\r\nSuppose we know that [latex]{a}^{3}=8[\/latex]. We want to find what number raised to the 3rd power is equal to 8. Since [latex]{2}^{3}=8[\/latex], we say that 2 is the cube root of 8. In the next example we will evaluate the cube roots of some perfect cubes.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nEvaluate the following:\r\n<ol>\r\n \t<li>[latex] \\sqrt[3]{125}[\/latex]<\/li>\r\n \t<li>[latex] \\sqrt[3]{-8}[\/latex]<\/li>\r\n \t<li>[latex] \\sqrt[3]{27}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"517592\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"517592\"]\r\n\r\n1. You can read this as \u201cthe third root of 125\u201d or \u201cthe cube root of 125.\u201d To evaluate this expression, look for a number that, when multiplied by itself two times (for a total of three identical factors), equals 125. [latex]\\text{?}\\cdot\\text{?}\\cdot\\text{?}=125[\/latex]. Since 125 ends in 5, 5 is a good candidate. [latex]5\\cdot5\\cdot5=125[\/latex]\r\n2. We want to find a number whose cube is 8. [latex]2\\cdot2\\cdot2=8[\/latex] the cube root of 8 is 2.\r\n\r\n3. We want to find a number whose cube is [latex]-8[\/latex]. We know [latex]2[\/latex] is the cube root of [latex]8[\/latex], so maybe we can try [latex]-2[\/latex]. [latex](-2)\\cdot{(-2)}\\cdot{(-2)}=-8[\/latex], so the cube root of [latex]-8[\/latex] is [latex]-2[\/latex]. This is different from square roots because multiplying three negative numbers together results in a negative number.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nAs we saw in the last example,there is one interesting fact about cube roots that is not true of square roots. Negative numbers can\u2019t have real number square roots, but negative numbers can have real number cube roots! What is the cube root of [latex]\u22128[\/latex]? [latex] \\sqrt[3]{-8}=-2[\/latex] because [latex] (-2)\\cdot (-2)\\cdot (-2)=-8[\/latex]. Remember, when you are multiplying an odd number of negative numbers, the result is negative! Consider [latex] \\sqrt[3]{{{(-1)}^{3}}}=-1[\/latex].\r\n\r\nIn the following video we show more examples of finding a cube root.\r\n\r\nhttps:\/\/youtu.be\/9Nh-Ggd2VJo\r\n<h1>(9.1.2) - Define and evaluate n<sup>th<\/sup> roots<\/h1>\r\nThe cube root of a number is written with a small number 3, called the <strong>index<\/strong>, just outside and above the radical symbol. It looks like [latex]\\displaystyle {\\,}^3\\hspace{-0.1in}\\sqrt{\\,\\,\\,}[\/latex]. This little 3 distinguishes cube roots from square roots which are written without a small number outside and above the radical symbol.\r\n\r\nWe can apply the same idea to any exponent and it's corresponding root.\u00a0 The [latex]n^{th}[\/latex]\u00a0root of [latex]a[\/latex] is a number that, when raised to the [latex]n^{th}[\/latex] 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 [latex]n^{th}[\/latex]\u00a0root, then the <strong>principal [latex]\\boldsymbol{n^{th}}[\/latex]\u00a0root<\/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 [latex]n^{th}[\/latex]\u00a0root of [latex]a[\/latex] is written as [latex]\\sqrt[n]{a}[\/latex], where [latex]n[\/latex] is a positive integer greater than or equal to 2. In the radical expression, [latex]n[\/latex] is called the <strong>index<\/strong> of the radical.\r\n<div class=\"textbox learning-objectives\">\r\n<h3>Definition:\u00a0Principal [latex]n^{th}[\/latex]\u00a0Root<\/h3>\r\nIf [latex]a[\/latex] is a real number with at least one [latex]n^{th}[\/latex]\u00a0root, then the <strong>principal [latex]\\boldsymbol{n^{th}}[\/latex]\u00a0root<\/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^{th}[\/latex]\u00a0power, equals [latex]a[\/latex]. The <strong>index<\/strong> of the radical is [latex]n[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nEvaluate each of the following:\r\n<ol>\r\n \t<li>[latex]\\sqrt[5]{-32}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt[4]{81}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt[8]{-1}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"140298\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"140298\"]\r\n<ol>\r\n \t<li>[latex]\\sqrt[5]{-32}[\/latex] Factor 32, because [latex]{\\left(-2\\right)}^{5}=-32 \\\\ \\text{ }[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt[4]{81}[\/latex]. Factoring can help, we know that [latex]9\\cdot9=81[\/latex] and we can further factor each 9: [latex]\\sqrt[4]{81}=\\sqrt[4]{3\\cdot3\\cdot3\\cdot3}=\\sqrt[4]{3^4}=3[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt[8]{-1}[\/latex], since we have an 8th root - which is even- with a negative number as the radicand, this root has no real number solutions. In other words, [latex](-1)\\cdot(-1)\\cdot(-1)\\cdot(-1)\\cdot(-1)\\cdot(-1)\\cdot(-1)\\cdot(-1)=+1[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video we show more examples of how to evaluate and nth root.\r\n\r\nhttps:\/\/youtu.be\/vA2DkcUSRSk\r\n\r\nYou can find the odd root of a negative number, but you cannot find the even root of a negative number. This means you can evaluate the radicals [latex] \\sqrt[3]{-81},\\ \\sqrt[5]{-64}[\/latex], and [latex] \\sqrt[7]{-2187}[\/latex], but you cannot evaluate the radicals [latex] \\sqrt[{}]{-100},\\ \\sqrt[4]{-16}[\/latex], or [latex] \\sqrt[6]{-2,500}[\/latex].\r\n<h1>(9.1.3) - Estimate roots that are not perfect<\/h1>\r\nAn approach to handling roots that are not perfect (squares, cubes, etc.)\u00a0 is to approximate them by comparing the values to perfect squares, cubes, or nth roots. Suppose you wanted to know the square root of 17. Let\u2019s look at how you might approximate it.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nEstimate. [latex] \\sqrt{17}[\/latex]\r\n\r\n[reveal-answer q=\"358591\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"358591\"]Think of two perfect squares that surround 17.\u00a017 is in between the perfect squares 16 and 25.\u00a0So, [latex] \\sqrt{17}[\/latex] must be in between [latex] \\sqrt{16}[\/latex] and [latex] \\sqrt{25}[\/latex].\r\n\r\nDetermine whether [latex] \\sqrt{17}[\/latex] is closer to 4 or to 5 and make another estimate.\r\n<p style=\"text-align: center\">[latex] \\sqrt{16}=4[\/latex] and [latex] \\sqrt{25}=5[\/latex]<\/p>\r\nSince 17 is closer to 16 than 25, [latex] \\sqrt{17}[\/latex] is probably about 4.1 or 4.2.\r\n\r\nUse trial and error to get a better estimate of [latex] \\sqrt{17}[\/latex]. Try squaring incrementally greater numbers, beginning with 4.1, to find a good approximation for [latex] \\sqrt{17}[\/latex].\r\n<p style=\"text-align: center\">[latex]\\left(4.1\\right)^{2}[\/latex]<\/p>\r\n[latex]\\left(4.1\\right)^{2}[\/latex]\u00a0gives a closer estimate than [latex](4.2)^{2}[\/latex].\r\n<p style=\"text-align: center\">[latex]4.1\\cdot4.1=16.81\\\\4.2\\cdot4.2=17.64[\/latex]<\/p>\r\nContinue to use trial and error to get an even better estimate.\r\n<p style=\"text-align: center\">[latex]4.12\\cdot4.12=16.9744\\\\4.13\\cdot4.13=17.0569[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex] \\sqrt{17}\\approx 4.12[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThis approximation is pretty close. If you kept using this trial and error strategy you could continue to find the square root to the thousandths, ten-thousandths, and hundred-thousandths places, but eventually it would become too tedious to do by hand.\r\n\r\nFor this reason, when you need to find a more precise approximation of a square root, you should use a calculator. Most calculators have a square root key [latex] (\\sqrt{{\\,\\,\\,}})[\/latex] that will give you the square root approximation quickly. On a simple 4-function calculator, you would likely key in the number that you want to take the square root of and then press the square root key.\r\n\r\nTry to find [latex] \\sqrt{17}[\/latex] using your calculator. Note that you will not be able to get an \u201cexact\u201d answer because [latex] \\sqrt{17}[\/latex] is an irrational number, a number that cannot be expressed as a fraction, and the decimal never terminates or repeats. To nine decimal positions, [latex] \\sqrt{17}[\/latex] is approximated as 4.123105626. A calculator can save a lot of time and yield a more precise square root when you are dealing with numbers that aren\u2019t perfect squares.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nApproximate [latex] \\sqrt[3]{30}[\/latex] and also find its value using a calculator.\r\n\r\n[reveal-answer q=\"71092\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"71092\"]Find the cubes that surround 30.\r\n\r\n30 is inbetween the perfect cubes 27 and 81.\r\n\r\n[latex] \\sqrt[3]{27}=3[\/latex] and [latex] \\sqrt[3]{81}=4[\/latex], so [latex] \\sqrt[3]{30}[\/latex] is between 3 and 4.\r\nUse a calculator.\r\n<p style=\"text-align: center\">[latex]\\sqrt[3]{30}\\approx3.10723[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\nBy approximation: [latex]3\\ge\\sqrt[3]{30}\\le4[\/latex]\r\n\r\nUsing a calculator: [latex] \\sqrt[3]{30}\\approx3.10723[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe following video shows another example of how to estimate a square root.\r\n\r\nhttps:\/\/youtu.be\/iNfalyW7olk","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>(9.1.1) &#8211; Define and evaluate principal square roots\n<ul>\n<li>Square roots<\/li>\n<li>Cube roots<\/li>\n<\/ul>\n<\/li>\n<li>(9.1.2) &#8211; Define and evaluate n<sup>th<\/sup> roots<\/li>\n<li>(9.1.3) &#8211; Estimate roots that are not perfect<\/li>\n<\/ul>\n<\/div>\n<h1>(9.1.1) &#8211; Define and evaluate principal square roots<\/h1>\n<p>Did you know that you can take the 6th root of a number? You have probably heard of a square root, written [latex]\\sqrt{}[\/latex], but you can also take a third, fourth and even a 5,000th root (if you really had to). In this lesson we will learn how a square root is defined and then we will build on that to form an understanding of nth roots. \u00a0We will use factoring and rules for exponents to simplify mathematical expressions that contain roots.<\/p>\n<p>The most common root is the <strong>square root<\/strong>. First, we will define what square roots are,\u00a0 and how you find the square root of a number. Then we will apply similar ideas to define and evaluate nth roots.<\/p>\n<p>Roots are the inverse of exponents, much like multiplication is the inverse of division. Recall\u00a0how exponents are defined, and written; with an exponent, as words, and as repeated multiplication.<\/p>\n<p><strong>Exponent:<\/strong> [latex]{{3}^{2}}[\/latex],\u00a0[latex]{{4}^{5}}[\/latex],\u00a0[latex]{{x}^{3}}[\/latex],\u00a0[latex]{{x}^{\\text{n}}}[\/latex]<\/p>\n<p><strong>Name:<\/strong>\u00a0\u201cThree squared\u201d or\u00a0\u201cThree to the second power\u201d,\u00a0\u201cFour to the fifth power\u201d,\u00a0\u201c<i>x<\/i> cubed\u201d,\u00a0\u201c<i>x<\/i> to the <i>n<\/i>th power\u201d<\/p>\n<p><strong>Repeated Multiplication:<\/strong>\u00a0[latex]3\\cdot 3[\/latex], \u00a0[latex]4\\cdot 4\\cdot 4\\cdot 4\\cdot 4[\/latex], \u00a0[latex]x\\cdot x\\cdot x[\/latex], \u00a0[latex]\\underbrace{x\\cdot x\\cdot x...\\cdot x}_{n\\text{ times}}[\/latex].<\/p>\n<p>Conversely,\u00a0 when you are trying to find the square root of a number (say, 25), you are trying to find a number that can be multiplied by itself to create that original number. In the case of 25, you can find that [latex]5\\cdot5=25[\/latex], so 5 must be the square root.<\/p>\n<h3>Square Roots<\/h3>\n<p>The symbol for the square root is called a <strong>radical symbol<\/strong> and looks like this: [latex]\\sqrt{\\,\\,\\,}[\/latex]. The expression [latex]\\sqrt{25}[\/latex] is read \u201cthe square root of twenty-five\u201d or \u201cradical twenty-five.\u201d The number that is written under the radical symbol is called the <strong>radicand<\/strong>.<br \/>\n<img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200220\/CNX_CAT_Figure_01_03_002.jpg\" alt=\"The expression: square root of twenty-five is enclosed in a circle. The circle has an arrow pointing to it labeled: Radical expression. The square root symbol has an arrow pointing to it labeled: Radical. The number twenty-five has an arrow pointing to it labeled: Radicand.\" \/><\/p>\n<p>The following table shows different radicals and their equivalent written and simplified forms.<\/p>\n<table style=\"width: 70%\">\n<thead>\n<tr>\n<th>Radical<\/th>\n<th>Name<\/th>\n<th>Simplified Form<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]\\sqrt{36}[\/latex]<\/td>\n<td>\u201cSquare root of thirty-six\u201d<\/p>\n<p>\u201cRadical thirty-six\u201d<\/td>\n<td>[latex]\\sqrt{36}=\\sqrt{6\\cdot 6}=6[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\sqrt{100}[\/latex]<\/td>\n<td>\u201cSquare root of one hundred\u201d<\/p>\n<p>\u201cRadical one hundred\u201d<\/td>\n<td>[latex]\\sqrt{100}=\\sqrt{10\\cdot 10}=10[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\sqrt{225}[\/latex]<\/td>\n<td>\u201cSquare root of two hundred twenty-five\u201d<\/p>\n<p>\u201cRadical two hundred twenty-five\u201d<\/td>\n<td>[latex]\\sqrt{225}=\\sqrt{15\\cdot 15}=15[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Consider [latex]\\sqrt{25}[\/latex] again. You may realize that there is another value that, when multiplied by itself, also results in 25. That number is [latex]\u22125[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}5\\cdot 5=25\\\\-5\\cdot -5=25\\end{array}[\/latex]<\/p>\n<p>By definition, the square root symbol always means to find the positive root, called the <strong>principal root<\/strong>. So while [latex]5\\cdot5[\/latex] and [latex]\u22125\\cdot\u22125[\/latex] both equal 25, only 5 is the principal root. You should also know that zero is special because it has only one square root: itself (since [latex]0\\cdot0=0[\/latex]).<\/p>\n<p>In our first example we will show you how to use radical notation to evaluate principal square roots.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Find the principal root of each expression.<\/p>\n<ol>\n<li>[latex]\\sqrt{100}[\/latex]<\/li>\n<li>[latex]\\sqrt{16}[\/latex]<\/li>\n<li>[latex]\\sqrt{25+144}[\/latex]<\/li>\n<li>[latex]\\sqrt{49}-\\sqrt{81}[\/latex]<\/li>\n<li>[latex]-\\sqrt{81}[\/latex]<\/li>\n<li>[latex]\\sqrt{-9}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q419579\">Show Answer<\/span><\/p>\n<div id=\"q419579\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]\\sqrt{100}=10[\/latex] because [latex]{10}^{2}=100[\/latex]<\/li>\n<li>[latex]\\sqrt{\\sqrt{16}}=\\sqrt{4}=2[\/latex] because [latex]{4}^{2}=16[\/latex] and [latex]{2}^{2}=4[\/latex]<\/li>\n<li>Recall that square roots act as grouping symbols in the order of operations, so addition and subtraction must be performed first when they occur under a radical. [latex]\\sqrt{25+144}=\\sqrt{169}=13[\/latex] because [latex]{13}^{2}=169[\/latex]<\/li>\n<li>This problem is similar to the last one, but this time subtraction should occur after evaluating the root. Stop and think about why these two problems are different. [latex]\\sqrt{49}-\\sqrt{81}=7 - 9=-2[\/latex] because [latex]{7}^{2}=49[\/latex] and [latex]{9}^{2}=81[\/latex]<\/li>\n<li>\n<p style=\"text-align: left\">The negative in front means to take the opposite of the value after you simplify the radical. [latex]-\\sqrt{81} = -\\sqrt{9\\cdot 9}[\/latex].\u00a0 The square root of [latex]81[\/latex] is [latex]9[\/latex]. Then, take the opposite of [latex]9:\u00a0 [latex]\u2212(9)[\/latex]<\/p>\n<\/li>\n<li>[latex]\\sqrt{-9}[\/latex], we are looking for a number that when it is squared, returns [latex]-9[\/latex]. We can try [latex](-3)^2[\/latex], but that will give a positive result, and [latex]3^2[\/latex] will also give a positive result. This leads to an important fact - \u00a0you cannot find the square root of a negative number.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video we present more examples of how to find a principle square root.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Simplify a Variety of Square Expressions (Simplify Perfectly)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/2cWAkmJoaDQ?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>The last example we showed leads to an important characteristic of square roots. You can only take the square root of values that are nonnegative.<\/p>\n<p class=\"textbox shaded\"><strong>Domain of a Square Root<\/strong><br \/>\n[latex]\\sqrt{-a}[\/latex] is not defined for all real numbers, a. Therefore, [latex]\\sqrt{a}[\/latex] is defined for [latex]a\\ge0[\/latex]<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Think About It<\/h3>\n<p>Does [latex]\\sqrt{25}=\\pm 5[\/latex]? Write your ideas and a sentence to defend them in the box below before you look at the answer.<\/p>\n<p><textarea aria-label=\"Your Answer\" rows=\"1\"><\/textarea><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q101071\">Show Answer<\/span><\/p>\n<div id=\"q101071\" class=\"hidden-answer\" style=\"display: none\">\n<p><em>No. Although both<\/em> [latex]{5}^{2}[\/latex] <em>and<\/em> [latex]{\\left(-5\\right)}^{2}[\/latex] <em>are<\/em> [latex]25[\/latex], <em>the radical symbol implies only a nonnegative root, the principal square root. The principal square root of 25 is<\/em> [latex]\\sqrt{25}=5[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h3>Cube Roots<\/h3>\n<p>We 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.<\/p>\n<p>Suppose we know that [latex]{a}^{3}=8[\/latex]. We want to find what number raised to the 3rd power is equal to 8. Since [latex]{2}^{3}=8[\/latex], we say that 2 is the cube root of 8. In the next example we will evaluate the cube roots of some perfect cubes.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Evaluate the following:<\/p>\n<ol>\n<li>[latex]\\sqrt[3]{125}[\/latex]<\/li>\n<li>[latex]\\sqrt[3]{-8}[\/latex]<\/li>\n<li>[latex]\\sqrt[3]{27}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q517592\">Show Solution<\/span><\/p>\n<div id=\"q517592\" class=\"hidden-answer\" style=\"display: none\">\n<p>1. You can read this as \u201cthe third root of 125\u201d or \u201cthe cube root of 125.\u201d To evaluate this expression, look for a number that, when multiplied by itself two times (for a total of three identical factors), equals 125. [latex]\\text{?}\\cdot\\text{?}\\cdot\\text{?}=125[\/latex]. Since 125 ends in 5, 5 is a good candidate. [latex]5\\cdot5\\cdot5=125[\/latex]<br \/>\n2. We want to find a number whose cube is 8. [latex]2\\cdot2\\cdot2=8[\/latex] the cube root of 8 is 2.<\/p>\n<p>3. We want to find a number whose cube is [latex]-8[\/latex]. We know [latex]2[\/latex] is the cube root of [latex]8[\/latex], so maybe we can try [latex]-2[\/latex]. [latex](-2)\\cdot{(-2)}\\cdot{(-2)}=-8[\/latex], so the cube root of [latex]-8[\/latex] is [latex]-2[\/latex]. This is different from square roots because multiplying three negative numbers together results in a negative number.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>As we saw in the last example,there is one interesting fact about cube roots that is not true of square roots. Negative numbers can\u2019t have real number square roots, but negative numbers can have real number cube roots! What is the cube root of [latex]\u22128[\/latex]? [latex]\\sqrt[3]{-8}=-2[\/latex] because [latex](-2)\\cdot (-2)\\cdot (-2)=-8[\/latex]. Remember, when you are multiplying an odd number of negative numbers, the result is negative! Consider [latex]\\sqrt[3]{{{(-1)}^{3}}}=-1[\/latex].<\/p>\n<p>In the following video we show more examples of finding a cube root.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Simplify Cube Roots (Perfect Cube Radicands)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/9Nh-Ggd2VJo?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h1>(9.1.2) - Define and evaluate n<sup>th<\/sup> roots<\/h1>\n<p>The cube root of a number is written with a small number 3, called the <strong>index<\/strong>, just outside and above the radical symbol. It looks like [latex]\\displaystyle {\\,}^3\\hspace{-0.1in}\\sqrt{\\,\\,\\,}[\/latex]. This little 3 distinguishes cube roots from square roots which are written without a small number outside and above the radical symbol.<\/p>\n<p>We can apply the same idea to any exponent and it's corresponding root.\u00a0 The [latex]n^{th}[\/latex]\u00a0root of [latex]a[\/latex] is a number that, when raised to the [latex]n^{th}[\/latex] 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 [latex]n^{th}[\/latex]\u00a0root, then the <strong>principal [latex]\\boldsymbol{n^{th}}[\/latex]\u00a0root<\/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 [latex]n^{th}[\/latex]\u00a0root of [latex]a[\/latex] is written as [latex]\\sqrt[n]{a}[\/latex], where [latex]n[\/latex] is a positive integer greater than or equal to 2. In the radical expression, [latex]n[\/latex] is called the <strong>index<\/strong> of the radical.<\/p>\n<div class=\"textbox learning-objectives\">\n<h3>Definition:\u00a0Principal [latex]n^{th}[\/latex]\u00a0Root<\/h3>\n<p>If [latex]a[\/latex] is a real number with at least one [latex]n^{th}[\/latex]\u00a0root, then the <strong>principal [latex]\\boldsymbol{n^{th}}[\/latex]\u00a0root<\/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^{th}[\/latex]\u00a0power, equals [latex]a[\/latex]. The <strong>index<\/strong> of the radical is [latex]n[\/latex].<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Evaluate each of the following:<\/p>\n<ol>\n<li>[latex]\\sqrt[5]{-32}[\/latex]<\/li>\n<li>[latex]\\sqrt[4]{81}[\/latex]<\/li>\n<li>[latex]\\sqrt[8]{-1}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q140298\">Show Answer<\/span><\/p>\n<div id=\"q140298\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]\\sqrt[5]{-32}[\/latex] Factor 32, because [latex]{\\left(-2\\right)}^{5}=-32 \\\\ \\text{ }[\/latex]<\/li>\n<li>[latex]\\sqrt[4]{81}[\/latex]. Factoring can help, we know that [latex]9\\cdot9=81[\/latex] and we can further factor each 9: [latex]\\sqrt[4]{81}=\\sqrt[4]{3\\cdot3\\cdot3\\cdot3}=\\sqrt[4]{3^4}=3[\/latex]<\/li>\n<li>[latex]\\sqrt[8]{-1}[\/latex], since we have an 8th root - which is even- with a negative number as the radicand, this root has no real number solutions. In other words, [latex](-1)\\cdot(-1)\\cdot(-1)\\cdot(-1)\\cdot(-1)\\cdot(-1)\\cdot(-1)\\cdot(-1)=+1[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video we show more examples of how to evaluate and nth root.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Simplify Perfect Nth Roots\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/vA2DkcUSRSk?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can find the odd root of a negative number, but you cannot find the even root of a negative number. This means you can evaluate the radicals [latex]\\sqrt[3]{-81},\\ \\sqrt[5]{-64}[\/latex], and [latex]\\sqrt[7]{-2187}[\/latex], but you cannot evaluate the radicals [latex]\\sqrt[{}]{-100},\\ \\sqrt[4]{-16}[\/latex], or [latex]\\sqrt[6]{-2,500}[\/latex].<\/p>\n<h1>(9.1.3) - Estimate roots that are not perfect<\/h1>\n<p>An approach to handling roots that are not perfect (squares, cubes, etc.)\u00a0 is to approximate them by comparing the values to perfect squares, cubes, or nth roots. Suppose you wanted to know the square root of 17. Let\u2019s look at how you might approximate it.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Estimate. [latex]\\sqrt{17}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q358591\">Show Solution<\/span><\/p>\n<div id=\"q358591\" class=\"hidden-answer\" style=\"display: none\">Think of two perfect squares that surround 17.\u00a017 is in between the perfect squares 16 and 25.\u00a0So, [latex]\\sqrt{17}[\/latex] must be in between [latex]\\sqrt{16}[\/latex] and [latex]\\sqrt{25}[\/latex].<\/p>\n<p>Determine whether [latex]\\sqrt{17}[\/latex] is closer to 4 or to 5 and make another estimate.<\/p>\n<p style=\"text-align: center\">[latex]\\sqrt{16}=4[\/latex] and [latex]\\sqrt{25}=5[\/latex]<\/p>\n<p>Since 17 is closer to 16 than 25, [latex]\\sqrt{17}[\/latex] is probably about 4.1 or 4.2.<\/p>\n<p>Use trial and error to get a better estimate of [latex]\\sqrt{17}[\/latex]. Try squaring incrementally greater numbers, beginning with 4.1, to find a good approximation for [latex]\\sqrt{17}[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\left(4.1\\right)^{2}[\/latex]<\/p>\n<p>[latex]\\left(4.1\\right)^{2}[\/latex]\u00a0gives a closer estimate than [latex](4.2)^{2}[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]4.1\\cdot4.1=16.81\\\\4.2\\cdot4.2=17.64[\/latex]<\/p>\n<p>Continue to use trial and error to get an even better estimate.<\/p>\n<p style=\"text-align: center\">[latex]4.12\\cdot4.12=16.9744\\\\4.13\\cdot4.13=17.0569[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\sqrt{17}\\approx 4.12[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>This approximation is pretty close. If you kept using this trial and error strategy you could continue to find the square root to the thousandths, ten-thousandths, and hundred-thousandths places, but eventually it would become too tedious to do by hand.<\/p>\n<p>For this reason, when you need to find a more precise approximation of a square root, you should use a calculator. Most calculators have a square root key [latex](\\sqrt{{\\,\\,\\,}})[\/latex] that will give you the square root approximation quickly. On a simple 4-function calculator, you would likely key in the number that you want to take the square root of and then press the square root key.<\/p>\n<p>Try to find [latex]\\sqrt{17}[\/latex] using your calculator. Note that you will not be able to get an \u201cexact\u201d answer because [latex]\\sqrt{17}[\/latex] is an irrational number, a number that cannot be expressed as a fraction, and the decimal never terminates or repeats. To nine decimal positions, [latex]\\sqrt{17}[\/latex] is approximated as 4.123105626. A calculator can save a lot of time and yield a more precise square root when you are dealing with numbers that aren\u2019t perfect squares.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Approximate [latex]\\sqrt[3]{30}[\/latex] and also find its value using a calculator.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q71092\">Show Solution<\/span><\/p>\n<div id=\"q71092\" class=\"hidden-answer\" style=\"display: none\">Find the cubes that surround 30.<\/p>\n<p>30 is inbetween the perfect cubes 27 and 81.<\/p>\n<p>[latex]\\sqrt[3]{27}=3[\/latex] and [latex]\\sqrt[3]{81}=4[\/latex], so [latex]\\sqrt[3]{30}[\/latex] is between 3 and 4.<br \/>\nUse a calculator.<\/p>\n<p style=\"text-align: center\">[latex]\\sqrt[3]{30}\\approx3.10723[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>By approximation: [latex]3\\ge\\sqrt[3]{30}\\le4[\/latex]<\/p>\n<p>Using a calculator: [latex]\\sqrt[3]{30}\\approx3.10723[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The following video shows another example of how to estimate a square root.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" title=\"Approximate a Square Root to Two Decimal Places Using Trial and Error\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/iNfalyW7olk?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n\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-5030\">\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>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 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