{"id":1116,"date":"2021-10-13T17:08:19","date_gmt":"2021-10-13T17:08:19","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/?post_type=chapter&#038;p=1116"},"modified":"2026-04-01T13:47:37","modified_gmt":"2026-04-01T13:47:37","slug":"1-2-5-square-roots-and-exponents","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/chapter\/1-2-5-square-roots-and-exponents\/","title":{"raw":"1.2.4: Exponents and Square Roots","rendered":"1.2.4: Exponents and Square Roots"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h1>Learning Outcomes<\/h1>\r\n<ul>\r\n \t<li>Evaluate exponential expressions<\/li>\r\n \t<li>Evaluate square root expressions<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h1>Key words<\/h1>\r\n<ul>\r\n \t<li><strong>Exponential expression<\/strong>: notation to write repeated multiplication<\/li>\r\n \t<li><strong>Base<\/strong>: the number being raised to a power in an exponential expression<\/li>\r\n \t<li><strong>Exponent<\/strong>: the power the base is being raised to in an exponential expression<\/li>\r\n \t<li><strong>Radical<\/strong>: the sign used to indicate square root: [latex]\\sqrt{}[\/latex]<\/li>\r\n \t<li><strong>Square root<\/strong>: a number when squared gives the number under the radical<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Exponential Expressions<\/h2>\r\nJust as multiplication is repeated addition, we use exponential notation to write repeated multiplication of the same quantity. For example, [latex]{2}^{4}[\/latex] means to multiply [latex]2[\/latex] by itself four times, So, [latex]{2}^{4}[\/latex] means [latex]2\\cdot 2\\cdot 2\\cdot 2[\/latex].\u00a0 Conversely, [latex]10\\cdot10\\cdot10[\/latex] can be written more succinctly as [latex]10^{3}[\/latex]. The [latex]10[\/latex] in [latex]10^{3}[\/latex]<sup>\u00a0<\/sup>is called the <em><b>base<\/b><\/em>. The [latex]3[\/latex] in [latex]10^{3}[\/latex]<sup>\u00a0<\/sup>is called the <em><b>exponent<\/b><\/em>. The expression [latex]10^{3}[\/latex] is called the <em><strong>exponential expression<\/strong><\/em>. Knowing the names for the parts of an exponential expression will help you learn how to perform mathematical operations on them.\r\n<div class=\"textbox shaded\">\r\n<h3>Exponential Notation<\/h3>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224353\/CNX_BMath_Figure_10_02_013_img.png\" alt=\"On the left side, a raised to the m is shown. The m is labeled in blue as an exponent. The a is labeled in red as the base. On the right, it says a to the m means multiply m factors of a. Below this, it says a to the m equals a times a times a times a, with m factors written below in blue.\" \/>\r\nThis is read [latex]a[\/latex] to the [latex]{m}^{\\mathrm{th}}[\/latex] power.\r\n\r\n<\/div>\r\nIn the expression [latex]{a}^{m}[\/latex], the exponent tells us how many times we multiply the base [latex]a[\/latex] by itself.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224355\/CNX_BMath_Figure_10_02_014_img.png\" alt=\"On the left side, 7 to the 3rd power is shown. Below is 7 times 7 times 7, with 3 factors written below. On the right side, parentheses negative 8 to the 5th power is shown. Below is negative 8 times negative 8 times negative 8 times negative 8 times negative 8, with 5 factors written below.\" \/>\r\n\r\n[latex]10^{3}[\/latex] is read as \u201c[latex]10[\/latex] to the third power\u201d or \u201c[latex]10[\/latex] cubed.\u201d It means [latex]10\\cdot10\\cdot10[\/latex], or [latex]1,000[\/latex].\r\n\r\n[latex]8^{2}[\/latex]\u00a0is read as \u201c[latex]8[\/latex] to the second power\u201d or \u201c[latex]8[\/latex] squared.\u201d It means [latex]8\\cdot8[\/latex], or [latex]64[\/latex].\r\n\r\n[latex]5^{4}[\/latex]\u00a0is read as \u201c[latex]5[\/latex] to the fourth power.\u201d It means [latex]5\\cdot5\\cdot5\\cdot5[\/latex], or [latex]625[\/latex].\r\n\r\nIf the exponential expression is negative, such as [latex]\u22123^{4}[\/latex], it means [latex]\u2013\\left(3\\cdot3\\cdot3\\cdot3\\right)[\/latex] or [latex]\u221281[\/latex].\r\n\r\nIf [latex]\u22123[\/latex] is to be the base, it must be written as [latex]\\left(\u22123\\right)^{4}[\/latex], which means [latex](\u22123)\\cdot(\u22123)\\cdot(\u22123)\\cdot(\u22123)=81[\/latex].\r\n\r\nYou can see that there is quite a difference, so you have to be very careful! The following examples show how to identify the base and the exponent, as well as how to identify the expanded and exponential format of writing repeated multiplication.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nIdentify the exponent and the base in the following terms, then simplify:\r\n<ol>\r\n \t<li>[latex]7^{2}[\/latex]<\/li>\r\n \t<li>[latex]2^{3}[\/latex]<\/li>\r\n \t<li>[latex]-5^{2}[\/latex]<\/li>\r\n \t<li>[latex]\\left(-5\\right)^{2}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"RB0002\"]Show Solution[\/reveal-answer]\r\n<p style=\"text-align: left;\">[hidden-answer a=\"RB0002\"]<\/p>\r\n1) [latex]7^{2}[\/latex]\r\n\r\nThe exponent is [latex]2[\/latex] and the base is [latex]7[\/latex]: \u00a0 [latex]7^{2}=7\\cdot{7}=49[\/latex]\r\n\r\n2) [latex]2^{3}[\/latex]\r\n\r\nThe exponent is [latex]3[\/latex], and the base is [latex]2[\/latex]: \u00a0 [latex]2^{3}=2\\cdot 2 \\cdot 2 =8[\/latex]\r\n\r\n3) \u00a0[latex]-5^{2}[\/latex]\r\n\r\nThe exponent is [latex]3[\/latex], and the base is [latex]5[\/latex], the [latex]-[\/latex] is not getting the exponent because there are no parentheses that tell us it is. \u00a0Thus simplified it is [latex]-\\left(5\\cdot 5\\cdot \\right)=-25[\/latex].\r\n\r\n4)\u00a0[latex]\\left(-5\\right)^{2}[\/latex]\r\n\r\nThe exponent is [latex]2[\/latex] and the base is [latex]-5[\/latex]: \u00a0 \u00a0 [latex]\\left(-5\\right)^{2}=-5\\cdot{-5}=25[\/latex]\r\n\r\n&nbsp;\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify:\r\n<ol>\r\n \t<li>[latex]{5}^{3}[\/latex]<\/li>\r\n \t<li>[latex]{9}^{1}[\/latex]<\/li>\r\n<\/ol>\r\nSolution\r\n<table id=\"eip-id1168469452397\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<th>1.<\/th>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]{5}^{3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply [latex]3[\/latex] factors of [latex]5[\/latex].<\/td>\r\n<td>[latex]5\\cdot 5\\cdot 5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]125[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168046009892\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<th>2.<\/th>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]{9}^{1}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply [latex]1[\/latex] factor of [latex]9[\/latex].<\/td>\r\n<td>[latex]9[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146094[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify:\r\n<ol>\r\n \t<li>[latex]{\\left(-3\\right)}^{4}[\/latex]<\/li>\r\n \t<li>[latex]{-3}^{4}[\/latex]\r\n[reveal-answer q=\"152453\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"152453\"]<\/li>\r\n<\/ol>\r\nSolution\r\n<table id=\"eip-id1168468562526\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<th>1.<\/th>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]{\\left(-3\\right)}^{4}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply four factors of [latex]\u22123[\/latex].<\/td>\r\n<td>[latex]\\left(-3\\right)\\left(-3\\right)\\left(-3\\right)\\left(-3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]81[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168048408997\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<th>2.<\/th>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]{-3}^{4}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply two factors.<\/td>\r\n<td>[latex]-\\left(3\\cdot 3\\cdot 3\\cdot 3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]-81[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNotice the similarities and differences in parts 1 and 2. Why are the answers different? In part 1 the parentheses tell us to raise the [latex](\u22123)[\/latex] to the [latex]4[\/latex]<sup>th<\/sup> power. In part 2 we raise only the [latex]3[\/latex] to the [latex]4[\/latex]<sup>th<\/sup> power and then find the opposite.\r\n\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]146097[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Simplify Expressions with Square Roots<\/h2>\r\nRemember that when a number [latex]n[\/latex] is multiplied by itself, we can write this as [latex]{n}^{2}[\/latex], which we read aloud as \"n squared\". For example, [latex]{8}^{2}[\/latex] is read as \"8 squared\".\r\n\r\nWe call [latex]64[\/latex] the <em>square<\/em> of [latex]8[\/latex] because [latex]{8}^{2}=64[\/latex]. Similarly, [latex]121[\/latex] is the square of [latex]11[\/latex], because [latex]{11}^{2}=121[\/latex].\r\n<div class=\"textbox shaded\">\r\n<h3>Square of a Number<\/h3>\r\nIf [latex]{n}^{2}=m[\/latex], then [latex]m[\/latex] is the square of [latex]n[\/latex].\r\n\r\n<\/div>\r\n<h2>Modeling Squares<\/h2>\r\nDo you know why we use the word <em>square<\/em>? If we construct a square with three tiles on each side, the total number of tiles would be nine.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221832\/CNX_BMath_Figure_05_07_001_img.png\" alt=\"A square is shown with 3 tiles on each side. There are a total of 9 tiles in the square.\" \/>\r\nThis is why we say that the square of three is nine.\r\n<p style=\"text-align: center;\">[latex]{3}^{2}=9[\/latex]<\/p>\r\nThe number [latex]9[\/latex] is called a <em><strong>perfect square<\/strong><\/em> because it is the square of a whole number.\r\n\r\nThe chart shows the squares of the natural numbers [latex]1[\/latex] through [latex]15[\/latex]. You can refer to it to help you identify the perfect squares.\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221834\/CNX_BMath_Figure_05_07_008_img.png\" alt=\"A table has two rows. The first row shows the values of n from n=1 to n=15. The second row shows the values of n square that are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, and 225.\" width=\"624\" height=\"59\" \/>\r\n<div class=\"textbox shaded\">\r\n<h3>Perfect Squares<\/h3>\r\nA perfect square is the square of a rational number.\r\n\r\n<\/div>\r\nWhat happens when you square a negative number?\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{cc}\\hfill {\\left(-8\\right)}^{2}&amp; =\\left(-8\\right)\\left(-8\\right)\\\\ &amp; =64\\hfill \\end{array}[\/latex]\r\nWhen we multiply two negative numbers, the product is always positive. So, the square of a negative number is always positive.<\/p>\r\nThe chart shows the squares of the negative integers from [latex]-1[\/latex] to [latex]-15[\/latex].\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221836\/CNX_BMath_Figure_05_07_009_img.png\" alt=\"A table has two rows. The first row shows the values of n from n=-1 to n=-15. The second row shows the values of n square that are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, and 225.\" width=\"624\" height=\"59\" \/>\r\nDid you notice that these squares are the same as the squares of the positive numbers?\r\n<h2>Square Roots<\/h2>\r\nSometimes we will need to look at the relationship between numbers and their squares in reverse. Because [latex]{10}^{2}=100[\/latex], we say [latex]100[\/latex] is the square of [latex]10[\/latex]. We can also say that [latex]10[\/latex] is a square root of [latex]100[\/latex].\r\n<div class=\"textbox shaded\">\r\n<h3>Square Root of a Number<\/h3>\r\nIf [latex]{n}^{2}=m[\/latex], then [latex]n[\/latex] is a square root of [latex]m[\/latex].\r\n\r\n<\/div>\r\nNotice [latex]{\\left(-10\\right)}^{2}=100[\/latex] also, so [latex]-10[\/latex] is also a square root of [latex]100[\/latex]. Therefore, both [latex]10[\/latex] and [latex]-10[\/latex] are square roots of [latex]100[\/latex].\r\n\r\nSo, every positive number has two square roots: one positive and one negative.\r\n\r\nWhat if we only want the positive square root of a positive number? The <em><strong>radical sign<\/strong>,<\/em> [latex]\\sqrt{\\phantom{0}}[\/latex], stands for the positive square root. The positive square root is also called the <em><strong>principal square root<\/strong><\/em>.\r\n<div class=\"textbox shaded\">\r\n<h3>Square Root Notation<\/h3>\r\n[latex]\\sqrt{m}[\/latex] is read as \"the square root of [latex]m\\text{.\"}[\/latex]\r\nIf [latex]m={n}^{2}[\/latex] then [latex]\\sqrt{m}=n[\/latex] for [latex]{n}\\ge 0[\/latex].\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221838\/CNX_BMath_Figure_05_07_003_img.png\" alt=\"A picture of an m inside a square root sign is shown. The sign is labeled as a radical sign and the m is labeled as the radicand.\" \/>\r\n\r\n<\/div>\r\nWe can also use the radical sign for the square root of zero. Because [latex]{0}^{2}=0,\\sqrt{0}=0[\/latex]. Notice that zero has only one square root.\r\nThe chart shows the square roots of the first [latex]15[\/latex] perfect square numbers.\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221840\/CNX_BMath_Figure_05_07_010_img.png\" alt=\"A table is shown with 2 rows. The first row contains the values: square root of 1, square root of 4, square root of 9, square root of 16, square root of 25, square root of 36, square root of 49, square root of 64, square root of 81, square root of 100, square root of 121, square root of 144, square root of 169, square root of 196, and square root of 225. The second row contains the values: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, and 15.\" width=\"598\" height=\"59\" \/>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: (a)\u00a0 [latex]\\sqrt{25}[\/latex] \u00a0\u00a0 (b)\u00a0 [latex]\\sqrt{121}[\/latex]\r\n\r\nSolution\r\n(a)\u00a0 [latex]\\sqrt{25}=5[\/latex] since [latex]{5}^{2}=25[\/latex]\r\n\r\n(b)\u00a0 [latex]\\sqrt{121}=11[\/latex] since [latex]{11}^{2}=121[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146618[\/ohm_question]\r\n\r\n<\/div>\r\nThe following video shows several more examples of how to simplify the square root of a perfect square.\r\n\r\nhttps:\/\/youtu.be\/rDpIm_EepcE\r\n\r\nEvery positive number has two square roots and the radical sign indicates the positive one. We write [latex]\\sqrt{100}=10[\/latex]. If we want to find the negative square root of a number, we place a negative in front of the radical sign. For example, [latex]-\\sqrt{100}=-10[\/latex].\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify. (a)\u00a0 [latex]-\\sqrt{9}[\/latex]\u00a0\u00a0 (b)\u00a0 [latex]-\\sqrt{144}[\/latex]\r\n[reveal-answer q=\"455014\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"455014\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168469874827\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<th>(a)<\/th>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-\\sqrt{9}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The negative is in front of the radical sign.<\/td>\r\n<td>[latex]-3[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168466190277\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<th>(b)<\/th>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-\\sqrt{144}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The negative is in front of the radical sign.<\/td>\r\n<td>[latex]-12[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\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]146619[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Square Root of a Negative Number<\/h2>\r\nCan we simplify [latex]\\sqrt{-25}?[\/latex] Is there a number whose square is [latex]-25?[\/latex]\r\n<p style=\"text-align: center;\">[latex]{\\left(?\\right)}^{2}=-25[\/latex]<\/p>\r\nNone of the numbers that we have dealt with so far have a square that is [latex]-25[\/latex]. Why? Any positive number squared is positive, and any negative number squared is also positive. We say there is no real number equal to [latex]\\sqrt{-25}[\/latex]. If we are asked to find the square root of any negative number, we say that the solution is not a real number.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: (a)\u00a0 [latex]\\sqrt{-169}[\/latex] \u00a0 (b)\u00a0 [latex]-\\sqrt{121}[\/latex].\r\n[reveal-answer q=\"288322\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"288322\"]\r\n\r\nSolution\r\n(a)\u00a0 There is no real number whose square is [latex]-169[\/latex]. Therefore, [latex]\\sqrt{-169}[\/latex] is not a real number.\r\n(b)\u00a0 The negative is in front of the radical sign, so we find the opposite of the square root of [latex]121[\/latex].\r\n<table id=\"eip-id1168466130843\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<th>(b)<\/th>\r\n<td>[latex]-\\sqrt{121}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The negative is in front of the radical.<\/td>\r\n<td>[latex]-11[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\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]146620[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h1>Learning Outcomes<\/h1>\n<ul>\n<li>Evaluate exponential expressions<\/li>\n<li>Evaluate square root expressions<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h1>Key words<\/h1>\n<ul>\n<li><strong>Exponential expression<\/strong>: notation to write repeated multiplication<\/li>\n<li><strong>Base<\/strong>: the number being raised to a power in an exponential expression<\/li>\n<li><strong>Exponent<\/strong>: the power the base is being raised to in an exponential expression<\/li>\n<li><strong>Radical<\/strong>: the sign used to indicate square root: [latex]\\sqrt{}[\/latex]<\/li>\n<li><strong>Square root<\/strong>: a number when squared gives the number under the radical<\/li>\n<\/ul>\n<\/div>\n<h2>Exponential Expressions<\/h2>\n<p>Just as multiplication is repeated addition, we use exponential notation to write repeated multiplication of the same quantity. For example, [latex]{2}^{4}[\/latex] means to multiply [latex]2[\/latex] by itself four times, So, [latex]{2}^{4}[\/latex] means [latex]2\\cdot 2\\cdot 2\\cdot 2[\/latex].\u00a0 Conversely, [latex]10\\cdot10\\cdot10[\/latex] can be written more succinctly as [latex]10^{3}[\/latex]. The [latex]10[\/latex] in [latex]10^{3}[\/latex]<sup>\u00a0<\/sup>is called the <em><b>base<\/b><\/em>. The [latex]3[\/latex] in [latex]10^{3}[\/latex]<sup>\u00a0<\/sup>is called the <em><b>exponent<\/b><\/em>. The expression [latex]10^{3}[\/latex] is called the <em><strong>exponential expression<\/strong><\/em>. Knowing the names for the parts of an exponential expression will help you learn how to perform mathematical operations on them.<\/p>\n<div class=\"textbox shaded\">\n<h3>Exponential Notation<\/h3>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224353\/CNX_BMath_Figure_10_02_013_img.png\" alt=\"On the left side, a raised to the m is shown. The m is labeled in blue as an exponent. The a is labeled in red as the base. On the right, it says a to the m means multiply m factors of a. Below this, it says a to the m equals a times a times a times a, with m factors written below in blue.\" \/><br \/>\nThis is read [latex]a[\/latex] to the [latex]{m}^{\\mathrm{th}}[\/latex] power.<\/p>\n<\/div>\n<p>In the expression [latex]{a}^{m}[\/latex], the exponent tells us how many times we multiply the base [latex]a[\/latex] by itself.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224355\/CNX_BMath_Figure_10_02_014_img.png\" alt=\"On the left side, 7 to the 3rd power is shown. Below is 7 times 7 times 7, with 3 factors written below. On the right side, parentheses negative 8 to the 5th power is shown. Below is negative 8 times negative 8 times negative 8 times negative 8 times negative 8, with 5 factors written below.\" \/><\/p>\n<p>[latex]10^{3}[\/latex] is read as \u201c[latex]10[\/latex] to the third power\u201d or \u201c[latex]10[\/latex] cubed.\u201d It means [latex]10\\cdot10\\cdot10[\/latex], or [latex]1,000[\/latex].<\/p>\n<p>[latex]8^{2}[\/latex]\u00a0is read as \u201c[latex]8[\/latex] to the second power\u201d or \u201c[latex]8[\/latex] squared.\u201d It means [latex]8\\cdot8[\/latex], or [latex]64[\/latex].<\/p>\n<p>[latex]5^{4}[\/latex]\u00a0is read as \u201c[latex]5[\/latex] to the fourth power.\u201d It means [latex]5\\cdot5\\cdot5\\cdot5[\/latex], or [latex]625[\/latex].<\/p>\n<p>If the exponential expression is negative, such as [latex]\u22123^{4}[\/latex], it means [latex]\u2013\\left(3\\cdot3\\cdot3\\cdot3\\right)[\/latex] or [latex]\u221281[\/latex].<\/p>\n<p>If [latex]\u22123[\/latex] is to be the base, it must be written as [latex]\\left(\u22123\\right)^{4}[\/latex], which means [latex](\u22123)\\cdot(\u22123)\\cdot(\u22123)\\cdot(\u22123)=81[\/latex].<\/p>\n<p>You can see that there is quite a difference, so you have to be very careful! The following examples show how to identify the base and the exponent, as well as how to identify the expanded and exponential format of writing repeated multiplication.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Identify the exponent and the base in the following terms, then simplify:<\/p>\n<ol>\n<li>[latex]7^{2}[\/latex]<\/li>\n<li>[latex]2^{3}[\/latex]<\/li>\n<li>[latex]-5^{2}[\/latex]<\/li>\n<li>[latex]\\left(-5\\right)^{2}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qRB0002\">Show Solution<\/span><\/p>\n<p style=\"text-align: left;\">\n<div id=\"qRB0002\" class=\"hidden-answer\" style=\"display: none\">\n<p>1) [latex]7^{2}[\/latex]<\/p>\n<p>The exponent is [latex]2[\/latex] and the base is [latex]7[\/latex]: \u00a0 [latex]7^{2}=7\\cdot{7}=49[\/latex]<\/p>\n<p>2) [latex]2^{3}[\/latex]<\/p>\n<p>The exponent is [latex]3[\/latex], and the base is [latex]2[\/latex]: \u00a0 [latex]2^{3}=2\\cdot 2 \\cdot 2 =8[\/latex]<\/p>\n<p>3) \u00a0[latex]-5^{2}[\/latex]<\/p>\n<p>The exponent is [latex]3[\/latex], and the base is [latex]5[\/latex], the [latex]-[\/latex] is not getting the exponent because there are no parentheses that tell us it is. \u00a0Thus simplified it is [latex]-\\left(5\\cdot 5\\cdot \\right)=-25[\/latex].<\/p>\n<p>4)\u00a0[latex]\\left(-5\\right)^{2}[\/latex]<\/p>\n<p>The exponent is [latex]2[\/latex] and the base is [latex]-5[\/latex]: \u00a0 \u00a0 [latex]\\left(-5\\right)^{2}=-5\\cdot{-5}=25[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify:<\/p>\n<ol>\n<li>[latex]{5}^{3}[\/latex]<\/li>\n<li>[latex]{9}^{1}[\/latex]<\/li>\n<\/ol>\n<p>Solution<\/p>\n<table id=\"eip-id1168469452397\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<th>1.<\/th>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]{5}^{3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply [latex]3[\/latex] factors of [latex]5[\/latex].<\/td>\n<td>[latex]5\\cdot 5\\cdot 5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]125[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168046009892\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<th>2.<\/th>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]{9}^{1}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply [latex]1[\/latex] factor of [latex]9[\/latex].<\/td>\n<td>[latex]9[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146094\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146094&theme=oea&iframe_resize_id=ohm146094&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify:<\/p>\n<ol>\n<li>[latex]{\\left(-3\\right)}^{4}[\/latex]<\/li>\n<li>[latex]{-3}^{4}[\/latex]\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q152453\">Show Solution<\/span><\/p>\n<div id=\"q152453\" class=\"hidden-answer\" style=\"display: none\"><\/li>\n<\/ol>\n<p>Solution<\/p>\n<table id=\"eip-id1168468562526\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<th>1.<\/th>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]{\\left(-3\\right)}^{4}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply four factors of [latex]\u22123[\/latex].<\/td>\n<td>[latex]\\left(-3\\right)\\left(-3\\right)\\left(-3\\right)\\left(-3\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]81[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168048408997\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<th>2.<\/th>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]{-3}^{4}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply two factors.<\/td>\n<td>[latex]-\\left(3\\cdot 3\\cdot 3\\cdot 3\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]-81[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Notice the similarities and differences in parts 1 and 2. Why are the answers different? In part 1 the parentheses tell us to raise the [latex](\u22123)[\/latex] to the [latex]4[\/latex]<sup>th<\/sup> power. In part 2 we raise only the [latex]3[\/latex] to the [latex]4[\/latex]<sup>th<\/sup> power and then find the opposite.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146097\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146097&theme=oea&iframe_resize_id=ohm146097&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Simplify Expressions with Square Roots<\/h2>\n<p>Remember that when a number [latex]n[\/latex] is multiplied by itself, we can write this as [latex]{n}^{2}[\/latex], which we read aloud as &#8220;n squared&#8221;. For example, [latex]{8}^{2}[\/latex] is read as &#8220;8 squared&#8221;.<\/p>\n<p>We call [latex]64[\/latex] the <em>square<\/em> of [latex]8[\/latex] because [latex]{8}^{2}=64[\/latex]. Similarly, [latex]121[\/latex] is the square of [latex]11[\/latex], because [latex]{11}^{2}=121[\/latex].<\/p>\n<div class=\"textbox shaded\">\n<h3>Square of a Number<\/h3>\n<p>If [latex]{n}^{2}=m[\/latex], then [latex]m[\/latex] is the square of [latex]n[\/latex].<\/p>\n<\/div>\n<h2>Modeling Squares<\/h2>\n<p>Do you know why we use the word <em>square<\/em>? If we construct a square with three tiles on each side, the total number of tiles would be nine.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221832\/CNX_BMath_Figure_05_07_001_img.png\" alt=\"A square is shown with 3 tiles on each side. There are a total of 9 tiles in the square.\" \/><br \/>\nThis is why we say that the square of three is nine.<\/p>\n<p style=\"text-align: center;\">[latex]{3}^{2}=9[\/latex]<\/p>\n<p>The number [latex]9[\/latex] is called a <em><strong>perfect square<\/strong><\/em> because it is the square of a whole number.<\/p>\n<p>The chart shows the squares of the natural numbers [latex]1[\/latex] through [latex]15[\/latex]. You can refer to it to help you identify the perfect squares.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221834\/CNX_BMath_Figure_05_07_008_img.png\" alt=\"A table has two rows. The first row shows the values of n from n=1 to n=15. The second row shows the values of n square that are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, and 225.\" width=\"624\" height=\"59\" \/><\/p>\n<div class=\"textbox shaded\">\n<h3>Perfect Squares<\/h3>\n<p>A perfect square is the square of a rational number.<\/p>\n<\/div>\n<p>What happens when you square a negative number?<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{cc}\\hfill {\\left(-8\\right)}^{2}& =\\left(-8\\right)\\left(-8\\right)\\\\ & =64\\hfill \\end{array}[\/latex]<br \/>\nWhen we multiply two negative numbers, the product is always positive. So, the square of a negative number is always positive.<\/p>\n<p>The chart shows the squares of the negative integers from [latex]-1[\/latex] to [latex]-15[\/latex].<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221836\/CNX_BMath_Figure_05_07_009_img.png\" alt=\"A table has two rows. The first row shows the values of n from n=-1 to n=-15. The second row shows the values of n square that are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, and 225.\" width=\"624\" height=\"59\" \/><br \/>\nDid you notice that these squares are the same as the squares of the positive numbers?<\/p>\n<h2>Square Roots<\/h2>\n<p>Sometimes we will need to look at the relationship between numbers and their squares in reverse. Because [latex]{10}^{2}=100[\/latex], we say [latex]100[\/latex] is the square of [latex]10[\/latex]. We can also say that [latex]10[\/latex] is a square root of [latex]100[\/latex].<\/p>\n<div class=\"textbox shaded\">\n<h3>Square Root of a Number<\/h3>\n<p>If [latex]{n}^{2}=m[\/latex], then [latex]n[\/latex] is a square root of [latex]m[\/latex].<\/p>\n<\/div>\n<p>Notice [latex]{\\left(-10\\right)}^{2}=100[\/latex] also, so [latex]-10[\/latex] is also a square root of [latex]100[\/latex]. Therefore, both [latex]10[\/latex] and [latex]-10[\/latex] are square roots of [latex]100[\/latex].<\/p>\n<p>So, every positive number has two square roots: one positive and one negative.<\/p>\n<p>What if we only want the positive square root of a positive number? The <em><strong>radical sign<\/strong>,<\/em> [latex]\\sqrt{\\phantom{0}}[\/latex], stands for the positive square root. The positive square root is also called the <em><strong>principal square root<\/strong><\/em>.<\/p>\n<div class=\"textbox shaded\">\n<h3>Square Root Notation<\/h3>\n<p>[latex]\\sqrt{m}[\/latex] is read as &#8220;the square root of [latex]m\\text{.\"}[\/latex]<br \/>\nIf [latex]m={n}^{2}[\/latex] then [latex]\\sqrt{m}=n[\/latex] for [latex]{n}\\ge 0[\/latex].<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221838\/CNX_BMath_Figure_05_07_003_img.png\" alt=\"A picture of an m inside a square root sign is shown. The sign is labeled as a radical sign and the m is labeled as the radicand.\" \/><\/p>\n<\/div>\n<p>We can also use the radical sign for the square root of zero. Because [latex]{0}^{2}=0,\\sqrt{0}=0[\/latex]. Notice that zero has only one square root.<br \/>\nThe chart shows the square roots of the first [latex]15[\/latex] perfect square numbers.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221840\/CNX_BMath_Figure_05_07_010_img.png\" alt=\"A table is shown with 2 rows. The first row contains the values: square root of 1, square root of 4, square root of 9, square root of 16, square root of 25, square root of 36, square root of 49, square root of 64, square root of 81, square root of 100, square root of 121, square root of 144, square root of 169, square root of 196, and square root of 225. The second row contains the values: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, and 15.\" width=\"598\" height=\"59\" \/><\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: (a)\u00a0 [latex]\\sqrt{25}[\/latex] \u00a0\u00a0 (b)\u00a0 [latex]\\sqrt{121}[\/latex]<\/p>\n<p>Solution<br \/>\n(a)\u00a0 [latex]\\sqrt{25}=5[\/latex] since [latex]{5}^{2}=25[\/latex]<\/p>\n<p>(b)\u00a0 [latex]\\sqrt{121}=11[\/latex] since [latex]{11}^{2}=121[\/latex]<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146618\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146618&theme=oea&iframe_resize_id=ohm146618&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>The following video shows several more examples of how to simplify the square root of a perfect square.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Simplify Square Roots (Perfect Squares)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/rDpIm_EepcE?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>Every positive number has two square roots and the radical sign indicates the positive one. We write [latex]\\sqrt{100}=10[\/latex]. If we want to find the negative square root of a number, we place a negative in front of the radical sign. For example, [latex]-\\sqrt{100}=-10[\/latex].<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify. (a)\u00a0 [latex]-\\sqrt{9}[\/latex]\u00a0\u00a0 (b)\u00a0 [latex]-\\sqrt{144}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q455014\">Show Solution<\/span><\/p>\n<div id=\"q455014\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168469874827\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<th>(a)<\/th>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]-\\sqrt{9}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The negative is in front of the radical sign.<\/td>\n<td>[latex]-3[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168466190277\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<th>(b)<\/th>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]-\\sqrt{144}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The negative is in front of the radical sign.<\/td>\n<td>[latex]-12[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146619\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146619&theme=oea&iframe_resize_id=ohm146619&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Square Root of a Negative Number<\/h2>\n<p>Can we simplify [latex]\\sqrt{-25}?[\/latex] Is there a number whose square is [latex]-25?[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]{\\left(?\\right)}^{2}=-25[\/latex]<\/p>\n<p>None of the numbers that we have dealt with so far have a square that is [latex]-25[\/latex]. Why? Any positive number squared is positive, and any negative number squared is also positive. We say there is no real number equal to [latex]\\sqrt{-25}[\/latex]. If we are asked to find the square root of any negative number, we say that the solution is not a real number.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: (a)\u00a0 [latex]\\sqrt{-169}[\/latex] \u00a0 (b)\u00a0 [latex]-\\sqrt{121}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q288322\">Show Solution<\/span><\/p>\n<div id=\"q288322\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<br \/>\n(a)\u00a0 There is no real number whose square is [latex]-169[\/latex]. Therefore, [latex]\\sqrt{-169}[\/latex] is not a real number.<br \/>\n(b)\u00a0 The negative is in front of the radical sign, so we find the opposite of the square root of [latex]121[\/latex].<\/p>\n<table id=\"eip-id1168466130843\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<th>(b)<\/th>\n<td>[latex]-\\sqrt{121}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The negative is in front of the radical.<\/td>\n<td>[latex]-11[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146620\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146620&theme=oea&iframe_resize_id=ohm146620&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\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-1116\">\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>Simplifying Square Roots (Perfect Squares). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/rDpIm_EepcE\">https:\/\/youtu.be\/rDpIm_EepcE<\/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>Adaption and Revision. <strong>Authored by<\/strong>: Roxanne Brinkerhoff. <strong>Provided by<\/strong>: Utah Valley University. <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 11: Exponents and Polynomials, from Developmental Math: An Open Program. <strong>Provided by<\/strong>: Monterey Institute of Technology and Education. <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":422605,"menu_order":6,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Unit 11: Exponents and Polynomials, from Developmental Math: An Open Program\",\"author\":\"\",\"organization\":\"Monterey Institute of Technology and Education\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Simplifying Square Roots (Perfect Squares)\",\"author\":\"James Sousa 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