{"id":648,"date":"2024-04-23T22:09:02","date_gmt":"2024-04-23T22:09:02","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=648"},"modified":"2025-08-13T14:53:31","modified_gmt":"2025-08-13T14:53:31","slug":"radicals-and-rational-exponents-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/radicals-and-rational-exponents-learn-it-1\/","title":{"raw":"Radicals and Rational Exponents: Learn It 1","rendered":"Radicals and Rational Exponents: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n \t<li>Calculate square roots and apply them in basic operations like addition, subtraction, and rationalizing the denominators.<\/li>\r\n \t<li>Use the product and quotient rules to simplify expressions that include square roots.<\/li>\r\n \t<li>Understand how to use rational exponents in expressions and recognize their connection to roots for simplifying calculations.<\/li>\r\n<\/ul>\r\n<\/section>\r\n\r\n[caption id=\"attachment_649\" align=\"alignright\" width=\"207\"]<img class=\"wp-image-649 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/04\/23220702\/CNX_CAT_Figure_01_03_001.jpg\" alt=\"\" width=\"207\" height=\"284\" \/> Triangle with two sides given[\/caption]\r\n\r\nA hardware store sells [latex]16[\/latex]-ft ladders and [latex]24[\/latex]-ft ladders. A window is located [latex]12[\/latex] feet above the ground. A ladder needs to be purchased that will reach the window from a point on the ground [latex]5[\/latex] feet from the building. To find out the length of ladder needed, we can draw a right triangle as shown, and use the Pythagorean Theorem.\r\n<p style=\"text-align: center;\">[latex]\\begin{align} {a}^{2}+{b}^{2}&amp; = {c}^{2} \\\\ {5}^{2}+{12}^{2}&amp; = {c}^{2} \\\\ 169&amp; = {c}^{2} \\end{align}[\/latex]<\/p>\r\nNow we need to find out the length that, when squared, is [latex]169,[\/latex] to determine which ladder to choose. In other word we need to find a square root. In this section we will investigate methods of finding solutions to problems such as this one.\r\n<h2>Square Roots<\/h2>\r\nWhen the square root of a number is squared, the result is the original number. Since [latex]{4}^{2}=16[\/latex], the square root of [latex]16[\/latex] is [latex]4[\/latex]. The square root function is the inverse of the squaring function just as subtraction is the inverse of addition. To undo squaring, we take the square root.\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>Square Root<\/h3>\r\nThe <strong>square root<\/strong> of a number [latex]a[\/latex] refers to any number [latex]x[\/latex] such that [latex]x^2 = a[\/latex].\r\n\r\nFor positive number [latex]a[\/latex], there are always two square roots: one positive and one negative.\r\n\r\n<\/div>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">For example, both [latex]+4[\/latex] and [latex]-4[\/latex] are square roots of [latex]16[\/latex] because both square to give [latex]16[\/latex].That is, [latex](+4)^2 = 16[\/latex] and [latex](-4)^2 = 16[\/latex].<\/section>When we talk about [pb_glossary id=\"651\"]square roots[\/pb_glossary] in math, we usually mean both the positive and negative numbers that, when multiplied by themselves, give us the original number. In math classes, especially when you're solving problems or learning algebra, it's important to think about both these answers unless we're told to find just one. This helps us understand all the possible solutions to an equation, which is a big part of learning how to solve math problems correctly.\r\n\r\nAs we explore square roots further, there's a special version we often use called the <strong>principal square root<\/strong>. This refers specifically to the positive square root of a number. It's helpful to know about this because it's commonly used in many situations.\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>principal square root<\/h3>\r\nThe <strong>principal square root<\/strong> of [latex]a[\/latex] is the nonnegative number that when multiplied by itself equals [latex]a[\/latex].\r\n\r\n&nbsp;\r\n\r\nThe principal square root of [latex]a[\/latex] is written as [latex]\\sqrt{a}[\/latex]. The symbol is called a <strong>radical<\/strong>, the term under the symbol is called the <strong>radicand<\/strong>, and the entire expression is called a <strong>radical expression<\/strong>.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/24203630\/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.\" width=\"487\" height=\"99\" \/> Items of the square root[\/caption]\r\n\r\n&nbsp;\r\n\r\nThe square root obtained using a calculator is the principal square root.\r\n\r\n<\/section><section class=\"textbox example\">Evaluate [latex]\\sqrt{25}[\/latex].[reveal-answer q=\"946328\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"946328\"]To evaluate [latex]\\sqrt{25}[\/latex], you are looking for the [pb_glossary id=\"653\"]principal square root[\/pb_glossary] of [latex]25[\/latex], which is the positive number that, when squared (multiplied by itself), equals [latex]25[\/latex].To find this:\r\n<ol>\r\n \t<li>Identify a number that when multiplied by itself gives [latex]25[\/latex].<\/li>\r\n \t<li>Since [latex]5 \\cdot 5 = 25[\/latex], and also [latex]-5 \\cdot -5 = 25[\/latex], there are two square roots: [latex]+5[\/latex] and [latex]-5[\/latex].<\/li>\r\n \t<li>However, we are referring to the principal square root, which is the positive value, when we are finding [latex]\\sqrt{25}[\/latex].<\/li>\r\n<\/ol>\r\nThus, [latex]\\sqrt{25} = 5[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox proTip\">The radical symbol of a number implies only a nonnegative root, the principal square root.<\/section><section class=\"textbox example\">Evaluate each expression.\r\n<ol>\r\n \t<li>[latex]\\sqrt{100}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{\\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<\/ol>\r\n[reveal-answer q=\"849035\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"849035\"]\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>[latex]\\sqrt{25+144}=\\sqrt{169}=13[\/latex] because [latex]{13}^{2}=169[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{49}-\\sqrt{81}=7 - 9=-2[\/latex] because [latex]{7}^{2}=49[\/latex] and [latex]{9}^{2}=81[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<strong>Question! <\/strong>For [latex]\\sqrt{25+144}[\/latex], can we find the square roots before adding?\r\n\r\n[reveal-answer q=\"705358\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"705358\"]<em><span style=\"text-decoration: underline;\"><strong>No!!<\/strong><\/span><\/em> If we take the square root of each number first: [latex]\\sqrt{25}+\\sqrt{144}=5+12=17[\/latex]. This is not equivalent to [latex]\\sqrt{25+144}=13[\/latex]. The [pb_glossary id=\"655\"]order of operations[\/pb_glossary] requires us to add the terms in the radicand before finding the square root.[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]18750[\/ohm2_question]<\/section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]18751[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Calculate square roots and apply them in basic operations like addition, subtraction, and rationalizing the denominators.<\/li>\n<li>Use the product and quotient rules to simplify expressions that include square roots.<\/li>\n<li>Understand how to use rational exponents in expressions and recognize their connection to roots for simplifying calculations.<\/li>\n<\/ul>\n<\/section>\n<figure id=\"attachment_649\" aria-describedby=\"caption-attachment-649\" style=\"width: 207px\" class=\"wp-caption alignright\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-649 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/04\/23220702\/CNX_CAT_Figure_01_03_001.jpg\" alt=\"\" width=\"207\" height=\"284\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/04\/23220702\/CNX_CAT_Figure_01_03_001.jpg 207w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/04\/23220702\/CNX_CAT_Figure_01_03_001-65x89.jpg 65w\" sizes=\"(max-width: 207px) 100vw, 207px\" \/><figcaption id=\"caption-attachment-649\" class=\"wp-caption-text\">Triangle with two sides given<\/figcaption><\/figure>\n<p>A hardware store sells [latex]16[\/latex]-ft ladders and [latex]24[\/latex]-ft ladders. A window is located [latex]12[\/latex] feet above the ground. A ladder needs to be purchased that will reach the window from a point on the ground [latex]5[\/latex] feet from the building. To find out the length of ladder needed, we can draw a right triangle as shown, and use the Pythagorean Theorem.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align} {a}^{2}+{b}^{2}& = {c}^{2} \\\\ {5}^{2}+{12}^{2}& = {c}^{2} \\\\ 169& = {c}^{2} \\end{align}[\/latex]<\/p>\n<p>Now we need to find out the length that, when squared, is [latex]169,[\/latex] to determine which ladder to choose. In other word we need to find a square root. In this section we will investigate methods of finding solutions to problems such as this one.<\/p>\n<h2>Square Roots<\/h2>\n<p>When the square root of a number is squared, the result is the original number. Since [latex]{4}^{2}=16[\/latex], the square root of [latex]16[\/latex] is [latex]4[\/latex]. The square root function is the inverse of the squaring function just as subtraction is the inverse of addition. To undo squaring, we take the square root.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>Square Root<\/h3>\n<p>The <strong>square root<\/strong> of a number [latex]a[\/latex] refers to any number [latex]x[\/latex] such that [latex]x^2 = a[\/latex].<\/p>\n<p>For positive number [latex]a[\/latex], there are always two square roots: one positive and one negative.<\/p>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">For example, both [latex]+4[\/latex] and [latex]-4[\/latex] are square roots of [latex]16[\/latex] because both square to give [latex]16[\/latex].That is, [latex](+4)^2 = 16[\/latex] and [latex](-4)^2 = 16[\/latex].<\/section>\n<p>When we talk about <a class=\"glossary-term\" aria-haspopup=\"dialog\" aria-describedby=\"definition\" href=\"#term_648_651\">square roots<\/a> in math, we usually mean both the positive and negative numbers that, when multiplied by themselves, give us the original number. In math classes, especially when you&#8217;re solving problems or learning algebra, it&#8217;s important to think about both these answers unless we&#8217;re told to find just one. This helps us understand all the possible solutions to an equation, which is a big part of learning how to solve math problems correctly.<\/p>\n<p>As we explore square roots further, there&#8217;s a special version we often use called the <strong>principal square root<\/strong>. This refers specifically to the positive square root of a number. It&#8217;s helpful to know about this because it&#8217;s commonly used in many situations.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>principal square root<\/h3>\n<p>The <strong>principal square root<\/strong> of [latex]a[\/latex] is the nonnegative number that when multiplied by itself equals [latex]a[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<p>The principal square root of [latex]a[\/latex] is written as [latex]\\sqrt{a}[\/latex]. The symbol is called a <strong>radical<\/strong>, the term under the symbol is called the <strong>radicand<\/strong>, and the entire expression is called a <strong>radical expression<\/strong>.<\/p>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/24203630\/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.\" width=\"487\" height=\"99\" \/><figcaption class=\"wp-caption-text\">Items of the square root<\/figcaption><\/figure>\n<p>&nbsp;<\/p>\n<p>The square root obtained using a calculator is the principal square root.<\/p>\n<\/section>\n<section class=\"textbox example\">Evaluate [latex]\\sqrt{25}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q946328\">Show Answer<\/button><\/p>\n<div id=\"q946328\" class=\"hidden-answer\" style=\"display: none\">To evaluate [latex]\\sqrt{25}[\/latex], you are looking for the <a class=\"glossary-term\" aria-haspopup=\"dialog\" aria-describedby=\"definition\" href=\"#term_648_653\">principal square root<\/a> of [latex]25[\/latex], which is the positive number that, when squared (multiplied by itself), equals [latex]25[\/latex].To find this:<\/p>\n<ol>\n<li>Identify a number that when multiplied by itself gives [latex]25[\/latex].<\/li>\n<li>Since [latex]5 \\cdot 5 = 25[\/latex], and also [latex]-5 \\cdot -5 = 25[\/latex], there are two square roots: [latex]+5[\/latex] and [latex]-5[\/latex].<\/li>\n<li>However, we are referring to the principal square root, which is the positive value, when we are finding [latex]\\sqrt{25}[\/latex].<\/li>\n<\/ol>\n<p>Thus, [latex]\\sqrt{25} = 5[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox proTip\">The radical symbol of a number implies only a nonnegative root, the principal square root.<\/section>\n<section class=\"textbox example\">Evaluate each expression.<\/p>\n<ol>\n<li>[latex]\\sqrt{100}[\/latex]<\/li>\n<li>[latex]\\sqrt{\\sqrt{16}}[\/latex]<\/li>\n<li>[latex]\\sqrt{25+144}[\/latex]<\/li>\n<li>[latex]\\sqrt{49}-\\sqrt{81}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q849035\">Show Solution<\/button><\/p>\n<div id=\"q849035\" 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>[latex]\\sqrt{25+144}=\\sqrt{169}=13[\/latex] because [latex]{13}^{2}=169[\/latex]<\/li>\n<li>[latex]\\sqrt{49}-\\sqrt{81}=7 - 9=-2[\/latex] because [latex]{7}^{2}=49[\/latex] and [latex]{9}^{2}=81[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p><strong>Question! <\/strong>For [latex]\\sqrt{25+144}[\/latex], can we find the square roots before adding?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q705358\">Show Answer<\/button><\/p>\n<div id=\"q705358\" class=\"hidden-answer\" style=\"display: none\"><em><span style=\"text-decoration: underline;\"><strong>No!!<\/strong><\/span><\/em> If we take the square root of each number first: [latex]\\sqrt{25}+\\sqrt{144}=5+12=17[\/latex]. This is not equivalent to [latex]\\sqrt{25+144}=13[\/latex]. The <a class=\"glossary-term\" aria-haspopup=\"dialog\" aria-describedby=\"definition\" href=\"#term_648_655\">order of operations<\/a> requires us to add the terms in the radicand before finding the square root.<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm18750\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=18750&theme=lumen&iframe_resize_id=ohm18750&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm18751\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=18751&theme=lumen&iframe_resize_id=ohm18751&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<div class=\"glossary\"><span class=\"screen-reader-text\" id=\"definition\">definition<\/span><template id=\"term_648_651\"><div class=\"glossary__definition\" role=\"dialog\" data-id=\"term_648_651\"><div tabindex=\"-1\"><p>A square root of a number is a value that, when multiplied by itself, gives the original number. <\/p>\n<\/div><button><span aria-hidden=\"true\">&times;<\/span><span class=\"screen-reader-text\">Close definition<\/span><\/button><\/div><\/template><template id=\"term_648_653\"><div class=\"glossary__definition\" role=\"dialog\" data-id=\"term_648_653\"><div tabindex=\"-1\"><p>The principal square root of a number [latex]a[\/latex] is defined as the nonnegative square root of [latex]a[\/latex].<\/p>\n<\/div><button><span aria-hidden=\"true\">&times;<\/span><span class=\"screen-reader-text\">Close definition<\/span><\/button><\/div><\/template><template id=\"term_648_655\"><div class=\"glossary__definition\" role=\"dialog\" data-id=\"term_648_655\"><div tabindex=\"-1\"><p>The order of operations, abbreviated as PEMDAS, guides the sequence for solving math expressions:<br \/>\nParentheses<br \/>\nExponents<br \/>\nMultiplication and Division<br \/>\nAddition and Subtraction<\/p>\n<\/div><button><span aria-hidden=\"true\">&times;<\/span><span class=\"screen-reader-text\">Close definition<\/span><\/button><\/div><\/template><\/div>","protected":false},"author":12,"menu_order":22,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":32,"module-header":"learn_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/648"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/users\/12"}],"version-history":[{"count":14,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/648\/revisions"}],"predecessor-version":[{"id":7571,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/648\/revisions\/7571"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/parts\/32"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/648\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=648"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=648"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=648"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=648"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}