{"id":4799,"date":"2017-06-07T18:57:26","date_gmt":"2017-06-07T18:57:26","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/chapter\/read-square-roots\/"},"modified":"2017-08-15T12:00:57","modified_gmt":"2017-08-15T12:00:57","slug":"read-square-roots","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/chapter\/read-square-roots\/","title":{"raw":"Square Roots","rendered":"Square Roots"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Define square root<\/li>\r\n \t<li>Find square roots<\/li>\r\n<\/ul>\r\n<\/div>\r\nWe know how to square a number:\r\n\r\n[latex]5^2=25[\/latex] and [latex]\\left(-5\\right)^2=25[\/latex]\r\n\r\nTaking a square root is the opposite of squaring so we can make these statements:\r\n<ul>\r\n \t<li>5 is the nonngeative square root of 25<\/li>\r\n \t<li>-5 is the negative square root of 25<\/li>\r\n<\/ul>\r\nFind the square roots of the following numbers:\r\n<ol>\r\n \t<li>36<\/li>\r\n \t<li>81<\/li>\r\n \t<li>-49<\/li>\r\n \t<li>0<\/li>\r\n<\/ol>\r\n<ol>\r\n \t<li>We want to find a number whose square is 36. [latex]6^2=36[\/latex] therefore, \u00a0the nonnegative square root of 36 is 6 and the negative square root of 36 is -6<\/li>\r\n \t<li>We want to find a number whose square is 81. [latex]9^2=81[\/latex] therefore, \u00a0the nonnegative square root of 81 is 9 and the negative square root of 81 is -9<\/li>\r\n \t<li>We want to find a number whose square is -49. When you square a real number, the result is always positive. Stop and think about that for a second.\u00a0A negative number times itself is positive, and a positive number times itself is positive. \u00a0Therefore, -49 does not have square roots, there are no real number solutions to this question.<\/li>\r\n \t<li>We want to find a number whose square is 0. [latex]0^2=0[\/latex] therefore, \u00a0the nonnegative square root of 0 is 0. \u00a0We do not assign 0 a sign, so it has only one square root, and that is 0.<\/li>\r\n<\/ol>\r\nThe notation that we use to express a square root for any real number, a, is as follows:\r\n<div class=\"textbox shaded\">\r\n<h4>Writing a Square Root<\/h4>\r\nThe symbol for the square root is called a <strong>radical symbol.<\/strong>\u00a0For a real number, <em>a<\/em> the square root of <em>a<\/em> is written as [latex]\\sqrt{a}[\/latex]\r\n\r\nThe number that is written under the radical symbol is called the <strong>radicand<\/strong>.\r\n\r\nBy definition, the square root symbol, [latex]\\sqrt{\\hphantom{5}}[\/latex] always means to find the nonnegative\u00a0root, called the <strong>principal root<\/strong>.\r\n\r\n[latex]\\sqrt{-a}[\/latex] is not defined, therefore [latex]\\sqrt{a}[\/latex] is defined for [latex]a&gt;0[\/latex]\r\n\r\n<\/div>\r\nLet's do an example similar to\u00a0the example from above, this time using square root notation. \u00a0Note that using the square root notation means that you are only finding the principal root - the nonnegative root.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSimplify\u00a0the following square roots:\r\n<ol>\r\n \t<li>[latex]\\sqrt{16}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{9}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{-9}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{5^2}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"614386\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"614386\"]\r\n<ol>\r\n \t<li>[latex]\\sqrt{16}[\/latex]. \u00a0We are looking for a number whose square is 16, so\u00a0[latex]\\sqrt{16}=4[\/latex]. We only write the nonnegative root because that is how the root symbol is defined.<\/li>\r\n \t<li>[latex]\\sqrt{9}[\/latex]. \u00a0We are looking for a number whose square is 9, so [latex]\\sqrt{9}=3[\/latex].\u00a0We only write the nonnegative root because that is how the root symbol is defined.<\/li>\r\n \t<li>[latex]\\sqrt{-9}[\/latex]. We are looking for a number whose square is -9. \u00a0There are no real numbers whose square is -9, so this radical is not a real number.<\/li>\r\n \t<li>[latex]\\sqrt{5^2}[\/latex]. We are looking for a number whose square is [latex]5^2[\/latex]. \u00a0We already have the number whose square is [latex]5^2[\/latex], it's 5!<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe last problem in the previous example shows us an important relationship between squares and square roots, and we can summarize it as follows:\r\n<div class=\"textbox shaded\">\r\n<h4>\u00a0The square root of a square<\/h4>\r\nFor a nonnegative real number, a, [latex]\\sqrt{a^2}=a[\/latex]\r\n\r\n<\/div>\r\nIn the video that follows, we simplify\u00a0more square roots using the fact that\u00a0\u00a0[latex]\\sqrt{a^2}=a[\/latex] means finding the principal square root.\r\n\r\nhttps:\/\/youtu.be\/B3riJsl7uZM\r\n\r\n&nbsp;\r\n<h2>Summary<\/h2>\r\nThe square root of a number is the number which, when multiplied by itself, gives the original number. Principal square roots are always positive and the square root of 0 is 0. You can only take the square root of values that are nonnegative. The square root of a perfect square will be an integer.","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Define square root<\/li>\n<li>Find square roots<\/li>\n<\/ul>\n<\/div>\n<p>We know how to square a number:<\/p>\n<p>[latex]5^2=25[\/latex] and [latex]\\left(-5\\right)^2=25[\/latex]<\/p>\n<p>Taking a square root is the opposite of squaring so we can make these statements:<\/p>\n<ul>\n<li>5 is the nonngeative square root of 25<\/li>\n<li>-5 is the negative square root of 25<\/li>\n<\/ul>\n<p>Find the square roots of the following numbers:<\/p>\n<ol>\n<li>36<\/li>\n<li>81<\/li>\n<li>-49<\/li>\n<li>0<\/li>\n<\/ol>\n<ol>\n<li>We want to find a number whose square is 36. [latex]6^2=36[\/latex] therefore, \u00a0the nonnegative square root of 36 is 6 and the negative square root of 36 is -6<\/li>\n<li>We want to find a number whose square is 81. [latex]9^2=81[\/latex] therefore, \u00a0the nonnegative square root of 81 is 9 and the negative square root of 81 is -9<\/li>\n<li>We want to find a number whose square is -49. When you square a real number, the result is always positive. Stop and think about that for a second.\u00a0A negative number times itself is positive, and a positive number times itself is positive. \u00a0Therefore, -49 does not have square roots, there are no real number solutions to this question.<\/li>\n<li>We want to find a number whose square is 0. [latex]0^2=0[\/latex] therefore, \u00a0the nonnegative square root of 0 is 0. \u00a0We do not assign 0 a sign, so it has only one square root, and that is 0.<\/li>\n<\/ol>\n<p>The notation that we use to express a square root for any real number, a, is as follows:<\/p>\n<div class=\"textbox shaded\">\n<h4>Writing a Square Root<\/h4>\n<p>The symbol for the square root is called a <strong>radical symbol.<\/strong>\u00a0For a real number, <em>a<\/em> the square root of <em>a<\/em> is written as [latex]\\sqrt{a}[\/latex]<\/p>\n<p>The number that is written under the radical symbol is called the <strong>radicand<\/strong>.<\/p>\n<p>By definition, the square root symbol, [latex]\\sqrt{\\hphantom{5}}[\/latex] always means to find the nonnegative\u00a0root, called the <strong>principal root<\/strong>.<\/p>\n<p>[latex]\\sqrt{-a}[\/latex] is not defined, therefore [latex]\\sqrt{a}[\/latex] is defined for [latex]a>0[\/latex]<\/p>\n<\/div>\n<p>Let&#8217;s do an example similar to\u00a0the example from above, this time using square root notation. \u00a0Note that using the square root notation means that you are only finding the principal root &#8211; the nonnegative root.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Simplify\u00a0the following square roots:<\/p>\n<ol>\n<li>[latex]\\sqrt{16}[\/latex]<\/li>\n<li>[latex]\\sqrt{9}[\/latex]<\/li>\n<li>[latex]\\sqrt{-9}[\/latex]<\/li>\n<li>[latex]\\sqrt{5^2}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q614386\">Show Solution<\/span><\/p>\n<div id=\"q614386\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]\\sqrt{16}[\/latex]. \u00a0We are looking for a number whose square is 16, so\u00a0[latex]\\sqrt{16}=4[\/latex]. We only write the nonnegative root because that is how the root symbol is defined.<\/li>\n<li>[latex]\\sqrt{9}[\/latex]. \u00a0We are looking for a number whose square is 9, so [latex]\\sqrt{9}=3[\/latex].\u00a0We only write the nonnegative root because that is how the root symbol is defined.<\/li>\n<li>[latex]\\sqrt{-9}[\/latex]. We are looking for a number whose square is -9. \u00a0There are no real numbers whose square is -9, so this radical is not a real number.<\/li>\n<li>[latex]\\sqrt{5^2}[\/latex]. We are looking for a number whose square is [latex]5^2[\/latex]. \u00a0We already have the number whose square is [latex]5^2[\/latex], it&#8217;s 5!<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>The last problem in the previous example shows us an important relationship between squares and square roots, and we can summarize it as follows:<\/p>\n<div class=\"textbox shaded\">\n<h4>\u00a0The square root of a square<\/h4>\n<p>For a nonnegative real number, a, [latex]\\sqrt{a^2}=a[\/latex]<\/p>\n<\/div>\n<p>In the video that follows, we simplify\u00a0more square roots using the fact that\u00a0\u00a0[latex]\\sqrt{a^2}=a[\/latex] means finding the principal square root.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Simplify Square Roots (Perfect Square Radicands)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/B3riJsl7uZM?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n<h2>Summary<\/h2>\n<p>The square root of a number is the number which, when multiplied by itself, gives the original number. Principal square roots are always positive and the square root of 0 is 0. You can only take the square root of values that are nonnegative. The square root of a perfect square will be an integer.<\/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-4799\">\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>Image: Shortcut this way.. <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 Square Roots (Perfect Square Radicands). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/B3riJsl7uZM\">https:\/\/youtu.be\/B3riJsl7uZM<\/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 Square Roots (Not Perfect Square Radicands). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/oRd7aBCsmfU\">https:\/\/youtu.be\/oRd7aBCsmfU<\/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\/dm-opentext\">http:\/\/nrocnetwork.org\/dm-opentext<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/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":17533,"menu_order":1,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Image: Shortcut this way.\",\"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\/dm-opentext\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Simplify Square Roots (Perfect Square Radicands)\",\"author\":\"James Sousa (Mathispower4u.com) for 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