{"id":16419,"date":"2019-10-03T15:21:08","date_gmt":"2019-10-03T15:21:08","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/read-simplify-square-roots-with-variables\/"},"modified":"2021-05-21T21:27:06","modified_gmt":"2021-05-21T21:27:06","slug":"read-simplify-square-roots-with-variables","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/cpcc-algebra-trig-I-support\/chapter\/read-simplify-square-roots-with-variables\/","title":{"raw":"Simplifying Square Roots with Variables","rendered":"Simplifying Square Roots with Variables"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Simplify square roots with variables<\/li>\r\n \t<li>Recognize that by definition [latex]\\sqrt{x^{2}}[\/latex]\u00a0is always nonnegative<\/li>\r\n<\/ul>\r\n<\/div>\r\n<strong>Radical expressions<\/strong> are expressions that contain radicals. Radical expressions come in many forms, from simple and familiar, such as[latex] \\sqrt{16}[\/latex], to quite complicated, as in [latex] \\sqrt[3]{250{{x}^{4}}y}[\/latex]. Using factoring, you can simplify these radical expressions, too.\r\n\r\n[caption id=\"attachment_5108\" align=\"aligncenter\" width=\"677\"]<img class=\" wp-image-5108\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/06\/23013136\/Screen-Shot-2016-06-22-at-6.31.08-PM-300x109.png\" alt=\"Radical: of or going to the root or origin; fundamental: a radical difference\" width=\"677\" height=\"246\" \/> Radical[\/caption]\r\n<h2 class=\"Subsectiontitleunderline\">Simplifying Square Roots<\/h2>\r\nRadical expressions will sometimes include variables as well as numbers. Consider the expression [latex] \\sqrt{9{{x}^{6}}}[\/latex]. Simplifying a radical expression with\u00a0variables is not as straightforward as the examples we have already shown with integers.\r\n\r\nConsider the expression [latex] \\sqrt{{{x}^{2}}}[\/latex]. This looks like it should be equal to <i>x<\/i>, right? Let\u2019s test some values for <i>x<\/i> and see what happens.\r\n\r\nIn the chart below, look along each row and determine whether the value of <i>x<\/i> is the same as the value of [latex] \\sqrt{{{x}^{2}}}[\/latex]. Where are they equal? Where are they not equal?\r\n\r\nAfter doing that for each row, look again and determine whether the value of [latex] \\sqrt{{{x}^{2}}}[\/latex] is the same as the value of [latex]\\left|x\\right|[\/latex].\r\n<table>\r\n<thead>\r\n<tr>\r\n<th>[latex]x[\/latex]<\/th>\r\n<th>[latex]x^{2}[\/latex]<\/th>\r\n<th>[latex]\\sqrt{x^{2}}[\/latex]<\/th>\r\n<th>[latex]\\left|x\\right|[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]\u22125[\/latex]<\/td>\r\n<td>[latex]25[\/latex]<\/td>\r\n<td>[latex]5[\/latex]<\/td>\r\n<td>[latex]5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\u22122[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]6[\/latex]<\/td>\r\n<td>[latex]36[\/latex]<\/td>\r\n<td>[latex]6[\/latex]<\/td>\r\n<td>[latex]6[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]10[\/latex]<\/td>\r\n<td>[latex]100[\/latex]<\/td>\r\n<td>[latex]10[\/latex]<\/td>\r\n<td>[latex]10[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNotice\u2014in cases where <i>x<\/i> is a negative number, [latex]\\sqrt{x^{2}}\\neq{x}[\/latex]! (This happens because the process of squaring the number loses the negative sign, since a negative times a negative is a positive.) However, in all cases [latex]\\sqrt{x^{2}}=\\left|x\\right|[\/latex].\u00a0You need to consider this fact when simplifying radicals that contain variables, because by definition [latex]\\sqrt{x^{2}}[\/latex]\u00a0is always nonnegative.\r\n<div class=\"textbox shaded\">\r\n<h3>Taking the Square Root of a Radical Expression<\/h3>\r\nWhen finding the square root of an expression that contains variables raised to a power, consider that [latex]\\sqrt{x^{2}}=\\left|x\\right|[\/latex].\r\n\r\nExamples: [latex]\\sqrt{9x^{2}}=3\\left|x\\right|[\/latex], and [latex]\\sqrt{16{{x}^{2}}{{y}^{2}}}=4\\left|xy\\right|[\/latex]\r\n\r\n<\/div>\r\nLet\u2019s try it.\r\nThe goal is to find factors under the radical that are perfect squares so that you can take their square root.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\sqrt{9{{x}^{6}}}[\/latex]\r\n\r\n[reveal-answer q=\"41297\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"41297\"]Factor to find identical pairs.\r\n\r\n[latex] \\sqrt{3\\cdot 3\\cdot {{x}^{3}}\\cdot {{x}^{3}}}[\/latex]\r\n\r\nRewrite the pairs as perfect squares, note how we use the power rule for exponents to simplify [latex]x^6[\/latex] into a square: [latex]{x^3}^2[\/latex]\r\n\r\n[latex] \\sqrt{{{3}^{2}}\\cdot {{\\left( {{x}^{3}} \\right)}^{2}}}[\/latex]\r\n\r\nSeparate into individual radicals.\r\n\r\n[latex] \\sqrt{{{3}^{2}}}\\cdot \\sqrt{{{\\left( {{x}^{3}} \\right)}^{2}}}[\/latex]\r\n\r\nSimplify, using the rule that [latex] \\sqrt{{{x}^{2}}}=\\left|x\\right|[\/latex].\r\n\r\n[latex] 3\\left|{{x}^{3}}\\right|[\/latex]\r\n<h4>Answer<\/h4>\r\n[latex] \\sqrt{9{{x}^{6}}}=3\\left|{{x}^{3}}\\right|[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nVariable factors with even exponents can be written as squares. In the example above, [latex] {{x}^{6}}={{x}^{3}}\\cdot{{x}^{3}}={\\left|x^3\\right|}^{2}[\/latex] and\r\n\r\n[latex] {{y}^{4}}={{y}^{2}}\\cdot{{y}^{2}}={\\left(|y^2\\right|)}^{2}[\/latex].\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]189413[\/ohm_question]\r\n\r\n<\/div>\r\nLet\u2019s try to simplify another radical expression.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\sqrt{100{{x}^{2}}{{y}^{4}}}[\/latex]\r\n\r\n[reveal-answer q=\"982628\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"982628\"]Separate factors; look for squared numbers and variables. Factor 100 into [latex]10\\cdot10[\/latex].\r\n<p style=\"text-align: center;\">[latex] \\sqrt{10\\cdot 10\\cdot {{x}^{2}}\\cdot {{y}^{4}}}[\/latex]<\/p>\r\nFactor [latex]y^{4}[\/latex]\u00a0into [latex]\\left(y^{2}\\right)^{2}[\/latex].\r\n<p style=\"text-align: center;\">[latex] \\sqrt{10\\cdot 10\\cdot {{x}^{2}}\\cdot {{({{y}^{2}})}^{2}}}[\/latex]<\/p>\r\nSeparate the squared factors into individual radicals.\r\n<p style=\"text-align: center;\">[latex] \\sqrt{{{10}^{2}}}\\cdot \\sqrt{{{x}^{2}}}\\cdot \\sqrt{{{({{y}^{2}})}^{2}}}[\/latex]<\/p>\r\nTake the square root of each radical . Since you do not know whether <i>x<\/i> is positive or negative, use [latex]\\left|x\\right|[\/latex]\u00a0to account for both possibilities, thereby guaranteeing that your answer will be positive.\r\n<p style=\"text-align: center;\">[latex]10\\cdot\\left|x\\right|\\cdot{y}^{2}[\/latex]<\/p>\r\nSimplify and multiply.\r\n<p style=\"text-align: center;\">[latex]10\\left|x\\right|y^{2}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex] \\sqrt{100{{x}^{2}}{{y}^{4}}}=10\\left| x \\right|{{y}^{2}}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nYou can check your answer by squaring it to be sure it equals [latex] 100{{x}^{2}}{{y}^{4}}[\/latex].\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\sqrt{49{{x}^{10}}{{y}^{8}}}[\/latex]\r\n\r\n[reveal-answer q=\"283065\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"283065\"]Look for squared numbers and variables. Factor 49 into [latex]7\\cdot7[\/latex], [latex]x^{10}[\/latex]\u00a0into [latex]x^{5}\\cdot{x}^{5}[\/latex], and [latex]y^{8}[\/latex]\u00a0into [latex]y^{4}\\cdot{y}^{4}[\/latex].\r\n\r\n[latex] \\sqrt{7\\cdot 7\\cdot {{x}^{5}}\\cdot{{x}^{5}}\\cdot{{y}^{4}}\\cdot{{y}^{4}}}[\/latex]\r\n\r\nRewrite the pairs as squares.\r\n\r\n[latex] \\sqrt{{{7}^{2}}\\cdot{{({{x}^{5}})}^{2}}\\cdot{{({{y}^{4}})}^{2}}}[\/latex]\r\n\r\nSeparate the squared factors into individual radicals.\r\n\r\n[latex] \\sqrt{7^2}\\cdot\\sqrt{({x^5})^2}\\cdot\\sqrt{({y^4})^2}[\/latex]\r\n\r\nTake the square root of each radical using the rule that [latex] \\sqrt{{{x}^{2}}}=\\left|x\\right|[\/latex].\r\n\r\n[latex] 7\\cdot\\left|{{x}^{5}}\\right|\\cdot{{y}^{4}}[\/latex]\r\n\r\nMultiply.\r\n\r\n[latex] 7\\left|{{x}^{5}}\\right|{{y}^{4}}[\/latex]\r\n<h4>Answer<\/h4>\r\n[latex] \\sqrt{49{{x}^{10}}{{y}^{8}}}=7\\left|{{x}^{5}}\\right|{{y}^{4}}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nYou find that the square root of [latex] 49{{x}^{10}}{{y}^{8}}[\/latex] is [latex]7\\left|{{x}^{5}}\\right|{{y}^{4}}[\/latex]. In order to check this calculation, you could square [latex]7\\left|{{x}^{5}}\\right|{{y}^{4}}[\/latex], hoping to arrive at [latex] 49{{x}^{10}}{{y}^{8}}[\/latex]. And, in fact, you would get this expression if you evaluated [latex] {\\left({7\\left|{{x}^{5}}\\right|{{y}^{4}}}\\right)^{2}}[\/latex].\r\n\r\nIn the video that follows we show several examples of simplifying radicals with variables.\r\n\r\nhttps:\/\/youtu.be\/q7LqsKPoAKo\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\sqrt{{{a}^{3}}{{b}^{5}}{{c}^{2}}}[\/latex]\r\n\r\n[reveal-answer q=\"141094\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"141094\"]Factor to find variables with even exponents.\r\n\r\n[latex] \\sqrt{{{a}^{2}}\\cdot a\\cdot {{b}^{4}}\\cdot{b}\\cdot{{c}^{2}}}[\/latex]\r\n\r\nRewrite [latex]b^{4}[\/latex]\u00a0as [latex]\\left(b^{2}\\right)^{2}[\/latex].\r\n\r\n[latex] \\sqrt{{{a}^{2}}\\cdot a\\cdot {{({{b}^{2}})}^{2}}\\cdot{ b}\\cdot{{c}^{2}}}[\/latex]\r\n\r\nSeparate the squared factors into individual radicals.\r\n\r\n[latex] \\sqrt{{{a}^{2}}}\\cdot\\sqrt{{{({{b}^{2}})}^{2}}}\\cdot\\sqrt{{{c}^{2}}}\\cdot \\sqrt{a\\cdot b}[\/latex]\r\n\r\nTake the square root of each radical. Remember that [latex] \\sqrt{{{a}^{2}}}=\\left| a \\right|[\/latex].\r\n\r\n[latex] \\left| a \\right|\\cdot {{b}^{2}}\\cdot\\left|{c}\\right|\\cdot\\sqrt{a\\cdot b}[\/latex]\r\n\r\nSimplify and multiply. The entire quantity [latex] a{{b}^{2}}c[\/latex] can be enclosed in the absolute value sign because [latex]b^2[\/latex] will be positive anyway.\r\n\r\n[latex] \\left| a{{b}^{2}}c \\right|\\sqrt{ab}[\/latex]\r\n<h4>Answer<\/h4>\r\n[latex] \\sqrt{{{a}^{3}}{{b}^{5}}{{c}^{2}}}=\\left| a{{b}^{2}}c\\right|\\sqrt{ab}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the next section, we will explore cube roots, and use the methods we have shown here to simplify them. Cube roots are unique from square roots in that it is possible to have a negative number under the root, such as [latex]\\sqrt[3]{-125}[\/latex].\r\n<h2>Contribute!<\/h2><div style=\"margin-bottom: 8px;\">Did you have an idea for improving this content? We\u2019d love your input.<\/div><a href=\"https:\/\/docs.google.com\/document\/d\/1AXb2O-170bOBJHIndTrOiEtjB_EPLe_vAdDNVE7H22c\" target=\"_blank\" style=\"font-size: 10pt; font-weight: 600; color: #077fab; text-decoration: none; border: 2px solid #077fab; border-radius: 7px; padding: 5px 25px; text-align: center; cursor: pointer; line-height: 1.5em;\">Improve this page<\/a><a style=\"margin-left: 16px;\" target=\"_blank\" href=\"https:\/\/docs.google.com\/document\/d\/1vy-T6DtTF-BbMfpVEI7VP_R7w2A4anzYZLXR8Pk4Fu4\">Learn More<\/a>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Simplify square roots with variables<\/li>\n<li>Recognize that by definition [latex]\\sqrt{x^{2}}[\/latex]\u00a0is always nonnegative<\/li>\n<\/ul>\n<\/div>\n<p><strong>Radical expressions<\/strong> are expressions that contain radicals. Radical expressions come in many forms, from simple and familiar, such as[latex]\\sqrt{16}[\/latex], to quite complicated, as in [latex]\\sqrt[3]{250{{x}^{4}}y}[\/latex]. Using factoring, you can simplify these radical expressions, too.<\/p>\n<div id=\"attachment_5108\" style=\"width: 687px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-5108\" class=\"wp-image-5108\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/06\/23013136\/Screen-Shot-2016-06-22-at-6.31.08-PM-300x109.png\" alt=\"Radical: of or going to the root or origin; fundamental: a radical difference\" width=\"677\" height=\"246\" \/><\/p>\n<p id=\"caption-attachment-5108\" class=\"wp-caption-text\">Radical<\/p>\n<\/div>\n<h2 class=\"Subsectiontitleunderline\">Simplifying Square Roots<\/h2>\n<p>Radical expressions will sometimes include variables as well as numbers. Consider the expression [latex]\\sqrt{9{{x}^{6}}}[\/latex]. Simplifying a radical expression with\u00a0variables is not as straightforward as the examples we have already shown with integers.<\/p>\n<p>Consider the expression [latex]\\sqrt{{{x}^{2}}}[\/latex]. This looks like it should be equal to <i>x<\/i>, right? Let\u2019s test some values for <i>x<\/i> and see what happens.<\/p>\n<p>In the chart below, look along each row and determine whether the value of <i>x<\/i> is the same as the value of [latex]\\sqrt{{{x}^{2}}}[\/latex]. Where are they equal? Where are they not equal?<\/p>\n<p>After doing that for each row, look again and determine whether the value of [latex]\\sqrt{{{x}^{2}}}[\/latex] is the same as the value of [latex]\\left|x\\right|[\/latex].<\/p>\n<table>\n<thead>\n<tr>\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]x^{2}[\/latex]<\/th>\n<th>[latex]\\sqrt{x^{2}}[\/latex]<\/th>\n<th>[latex]\\left|x\\right|[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]\u22125[\/latex]<\/td>\n<td>[latex]25[\/latex]<\/td>\n<td>[latex]5[\/latex]<\/td>\n<td>[latex]5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\u22122[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]6[\/latex]<\/td>\n<td>[latex]36[\/latex]<\/td>\n<td>[latex]6[\/latex]<\/td>\n<td>[latex]6[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]10[\/latex]<\/td>\n<td>[latex]100[\/latex]<\/td>\n<td>[latex]10[\/latex]<\/td>\n<td>[latex]10[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Notice\u2014in cases where <i>x<\/i> is a negative number, [latex]\\sqrt{x^{2}}\\neq{x}[\/latex]! (This happens because the process of squaring the number loses the negative sign, since a negative times a negative is a positive.) However, in all cases [latex]\\sqrt{x^{2}}=\\left|x\\right|[\/latex].\u00a0You need to consider this fact when simplifying radicals that contain variables, because by definition [latex]\\sqrt{x^{2}}[\/latex]\u00a0is always nonnegative.<\/p>\n<div class=\"textbox shaded\">\n<h3>Taking the Square Root of a Radical Expression<\/h3>\n<p>When finding the square root of an expression that contains variables raised to a power, consider that [latex]\\sqrt{x^{2}}=\\left|x\\right|[\/latex].<\/p>\n<p>Examples: [latex]\\sqrt{9x^{2}}=3\\left|x\\right|[\/latex], and [latex]\\sqrt{16{{x}^{2}}{{y}^{2}}}=4\\left|xy\\right|[\/latex]<\/p>\n<\/div>\n<p>Let\u2019s try it.<br \/>\nThe goal is to find factors under the radical that are perfect squares so that you can take their square root.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\sqrt{9{{x}^{6}}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q41297\">Show Solution<\/span><\/p>\n<div id=\"q41297\" class=\"hidden-answer\" style=\"display: none\">Factor to find identical pairs.<\/p>\n<p>[latex]\\sqrt{3\\cdot 3\\cdot {{x}^{3}}\\cdot {{x}^{3}}}[\/latex]<\/p>\n<p>Rewrite the pairs as perfect squares, note how we use the power rule for exponents to simplify [latex]x^6[\/latex] into a square: [latex]{x^3}^2[\/latex]<\/p>\n<p>[latex]\\sqrt{{{3}^{2}}\\cdot {{\\left( {{x}^{3}} \\right)}^{2}}}[\/latex]<\/p>\n<p>Separate into individual radicals.<\/p>\n<p>[latex]\\sqrt{{{3}^{2}}}\\cdot \\sqrt{{{\\left( {{x}^{3}} \\right)}^{2}}}[\/latex]<\/p>\n<p>Simplify, using the rule that [latex]\\sqrt{{{x}^{2}}}=\\left|x\\right|[\/latex].<\/p>\n<p>[latex]3\\left|{{x}^{3}}\\right|[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\sqrt{9{{x}^{6}}}=3\\left|{{x}^{3}}\\right|[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Variable factors with even exponents can be written as squares. In the example above, [latex]{{x}^{6}}={{x}^{3}}\\cdot{{x}^{3}}={\\left|x^3\\right|}^{2}[\/latex] and<\/p>\n<p>[latex]{{y}^{4}}={{y}^{2}}\\cdot{{y}^{2}}={\\left(|y^2\\right|)}^{2}[\/latex].<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm189413\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=189413&theme=oea&iframe_resize_id=ohm189413&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Let\u2019s try to simplify another radical expression.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\sqrt{100{{x}^{2}}{{y}^{4}}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q982628\">Show Solution<\/span><\/p>\n<div id=\"q982628\" class=\"hidden-answer\" style=\"display: none\">Separate factors; look for squared numbers and variables. Factor 100 into [latex]10\\cdot10[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt{10\\cdot 10\\cdot {{x}^{2}}\\cdot {{y}^{4}}}[\/latex]<\/p>\n<p>Factor [latex]y^{4}[\/latex]\u00a0into [latex]\\left(y^{2}\\right)^{2}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt{10\\cdot 10\\cdot {{x}^{2}}\\cdot {{({{y}^{2}})}^{2}}}[\/latex]<\/p>\n<p>Separate the squared factors into individual radicals.<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt{{{10}^{2}}}\\cdot \\sqrt{{{x}^{2}}}\\cdot \\sqrt{{{({{y}^{2}})}^{2}}}[\/latex]<\/p>\n<p>Take the square root of each radical . Since you do not know whether <i>x<\/i> is positive or negative, use [latex]\\left|x\\right|[\/latex]\u00a0to account for both possibilities, thereby guaranteeing that your answer will be positive.<\/p>\n<p style=\"text-align: center;\">[latex]10\\cdot\\left|x\\right|\\cdot{y}^{2}[\/latex]<\/p>\n<p>Simplify and multiply.<\/p>\n<p style=\"text-align: center;\">[latex]10\\left|x\\right|y^{2}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\sqrt{100{{x}^{2}}{{y}^{4}}}=10\\left| x \\right|{{y}^{2}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>You can check your answer by squaring it to be sure it equals [latex]100{{x}^{2}}{{y}^{4}}[\/latex].<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\sqrt{49{{x}^{10}}{{y}^{8}}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q283065\">Show Solution<\/span><\/p>\n<div id=\"q283065\" class=\"hidden-answer\" style=\"display: none\">Look for squared numbers and variables. Factor 49 into [latex]7\\cdot7[\/latex], [latex]x^{10}[\/latex]\u00a0into [latex]x^{5}\\cdot{x}^{5}[\/latex], and [latex]y^{8}[\/latex]\u00a0into [latex]y^{4}\\cdot{y}^{4}[\/latex].<\/p>\n<p>[latex]\\sqrt{7\\cdot 7\\cdot {{x}^{5}}\\cdot{{x}^{5}}\\cdot{{y}^{4}}\\cdot{{y}^{4}}}[\/latex]<\/p>\n<p>Rewrite the pairs as squares.<\/p>\n<p>[latex]\\sqrt{{{7}^{2}}\\cdot{{({{x}^{5}})}^{2}}\\cdot{{({{y}^{4}})}^{2}}}[\/latex]<\/p>\n<p>Separate the squared factors into individual radicals.<\/p>\n<p>[latex]\\sqrt{7^2}\\cdot\\sqrt{({x^5})^2}\\cdot\\sqrt{({y^4})^2}[\/latex]<\/p>\n<p>Take the square root of each radical using the rule that [latex]\\sqrt{{{x}^{2}}}=\\left|x\\right|[\/latex].<\/p>\n<p>[latex]7\\cdot\\left|{{x}^{5}}\\right|\\cdot{{y}^{4}}[\/latex]<\/p>\n<p>Multiply.<\/p>\n<p>[latex]7\\left|{{x}^{5}}\\right|{{y}^{4}}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\sqrt{49{{x}^{10}}{{y}^{8}}}=7\\left|{{x}^{5}}\\right|{{y}^{4}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>You find that the square root of [latex]49{{x}^{10}}{{y}^{8}}[\/latex] is [latex]7\\left|{{x}^{5}}\\right|{{y}^{4}}[\/latex]. In order to check this calculation, you could square [latex]7\\left|{{x}^{5}}\\right|{{y}^{4}}[\/latex], hoping to arrive at [latex]49{{x}^{10}}{{y}^{8}}[\/latex]. And, in fact, you would get this expression if you evaluated [latex]{\\left({7\\left|{{x}^{5}}\\right|{{y}^{4}}}\\right)^{2}}[\/latex].<\/p>\n<p>In the video that follows we show several examples of simplifying radicals with variables.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Simplify Square Roots with Variables\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/q7LqsKPoAKo?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\sqrt{{{a}^{3}}{{b}^{5}}{{c}^{2}}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q141094\">Show Solution<\/span><\/p>\n<div id=\"q141094\" class=\"hidden-answer\" style=\"display: none\">Factor to find variables with even exponents.<\/p>\n<p>[latex]\\sqrt{{{a}^{2}}\\cdot a\\cdot {{b}^{4}}\\cdot{b}\\cdot{{c}^{2}}}[\/latex]<\/p>\n<p>Rewrite [latex]b^{4}[\/latex]\u00a0as [latex]\\left(b^{2}\\right)^{2}[\/latex].<\/p>\n<p>[latex]\\sqrt{{{a}^{2}}\\cdot a\\cdot {{({{b}^{2}})}^{2}}\\cdot{ b}\\cdot{{c}^{2}}}[\/latex]<\/p>\n<p>Separate the squared factors into individual radicals.<\/p>\n<p>[latex]\\sqrt{{{a}^{2}}}\\cdot\\sqrt{{{({{b}^{2}})}^{2}}}\\cdot\\sqrt{{{c}^{2}}}\\cdot \\sqrt{a\\cdot b}[\/latex]<\/p>\n<p>Take the square root of each radical. Remember that [latex]\\sqrt{{{a}^{2}}}=\\left| a \\right|[\/latex].<\/p>\n<p>[latex]\\left| a \\right|\\cdot {{b}^{2}}\\cdot\\left|{c}\\right|\\cdot\\sqrt{a\\cdot b}[\/latex]<\/p>\n<p>Simplify and multiply. The entire quantity [latex]a{{b}^{2}}c[\/latex] can be enclosed in the absolute value sign because [latex]b^2[\/latex] will be positive anyway.<\/p>\n<p>[latex]\\left| a{{b}^{2}}c \\right|\\sqrt{ab}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\sqrt{{{a}^{3}}{{b}^{5}}{{c}^{2}}}=\\left| a{{b}^{2}}c\\right|\\sqrt{ab}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the next section, we will explore cube roots, and use the methods we have shown here to simplify them. Cube roots are unique from square roots in that it is possible to have a negative number under the root, such as [latex]\\sqrt[3]{-125}[\/latex].<\/p>\n<h2>Contribute!<\/h2>\n<div style=\"margin-bottom: 8px;\">Did you have an idea for improving this content? We\u2019d love your input.<\/div>\n<p><a href=\"https:\/\/docs.google.com\/document\/d\/1AXb2O-170bOBJHIndTrOiEtjB_EPLe_vAdDNVE7H22c\" target=\"_blank\" style=\"font-size: 10pt; font-weight: 600; color: #077fab; text-decoration: none; border: 2px solid #077fab; border-radius: 7px; padding: 5px 25px; text-align: center; cursor: pointer; line-height: 1.5em;\">Improve this page<\/a><a style=\"margin-left: 16px;\" target=\"_blank\" href=\"https:\/\/docs.google.com\/document\/d\/1vy-T6DtTF-BbMfpVEI7VP_R7w2A4anzYZLXR8Pk4Fu4\">Learn More<\/a><\/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-16419\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Screenshot: radical. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Simplify Square Roots with Variables. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/q7LqsKPoAKo\">https:\/\/youtu.be\/q7LqsKPoAKo<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Unit 16: Radical Expressions and Quadratic Equations, from Developmental Math: An Open Program. <strong>Provided by<\/strong>: Monterey Institute of Technology and Education. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/\">http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":169554,"menu_order":34,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Screenshot: radical\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Unit 16: Radical Expressions and Quadratic Equations, from Developmental Math: An Open Program\",\"author\":\"\",\"organization\":\"Monterey Institute of Technology and Education\",\"url\":\" http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Simplify Square Roots with Variables\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/q7LqsKPoAKo\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"06f4479c8f51472f98284003fdedb3ce, 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