{"id":3644,"date":"2020-01-28T01:37:04","date_gmt":"2020-01-28T01:37:04","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/mathforlibscoreq\/?post_type=chapter&#038;p=3644"},"modified":"2021-02-05T23:49:58","modified_gmt":"2021-02-05T23:49:58","slug":"simplifying-expressions-with-square-roots","status":"web-only","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/chapter\/simplifying-expressions-with-square-roots\/","title":{"raw":"Simplifying Expressions With Square Roots","rendered":"Simplifying Expressions With Square Roots"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Simplify expressions with square roots using the order of operations<\/li>\r\n \t<li>Simplify expressions with square roots that contain variables<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h3>Square Roots and the Order of Operations<\/h3>\r\nWhen using the order of operations to simplify an expression that has square roots, we treat the radical sign as a grouping symbol. We simplify any expressions under the radical sign before performing other operations.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: (a)\u00a0 [latex]\\sqrt{25}+\\sqrt{144}[\/latex]\u00a0 \u00a0 (b) [latex]\\sqrt{25+144}[\/latex]\r\n\r\nSolution\r\n<table id=\"eip-id1168466048150\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>(a) Use the order of operations.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\sqrt{25}+\\sqrt{144}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify each radical.<\/td>\r\n<td>[latex]5+12[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add.<\/td>\r\n<td>[latex]17[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168469785439\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>(b) Use the order of operations.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\sqrt{25+144}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add under the radical sign.<\/td>\r\n<td>[latex]\\sqrt{169}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]13[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146635[\/ohm_question]\r\n\r\n[ohm_question]146636[\/ohm_question]\r\n\r\n<\/div>\r\nNotice the different answers in parts (a) and (b) of the example above. It is important to follow the order of operations correctly. In (a), we took each square root first and then added them. In (b), we added under the radical sign first and then found the square root.\r\n<h2>Simplify Variable Expressions with Square Roots<\/h2>\r\nExpressions with square root that we have looked at so far have not had any variables. What happens when we have to find a square root of a variable expression?\r\n\r\nConsider [latex]\\sqrt{9{x}^{2}}[\/latex], where [latex]x\\ge 0[\/latex]. Can you think of an expression whose square is [latex]9{x}^{2}?[\/latex]\r\n<p style=\"text-align: center\">[latex]\\begin{array}{ccc}\\hfill {\\left(?\\right)}^{2}&amp; =&amp; 9{x}^{2}\\hfill \\\\ \\hfill {\\left(3x\\right)}^{2}&amp; =&amp; 9{x}^{2}\\text{ so }\\sqrt{9{x}^{2}}=3x\\hfill \\end{array}[\/latex]<\/p>\r\nWhen we use a variable in a square root expression, for our work, we will assume that the variable represents a non-negative number. In every example and exercise that follows, each variable in a square root expression is greater than or equal to zero.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]\\sqrt{{x}^{2}}[\/latex], where [latex]x\\ge 0[\/latex]\r\n[reveal-answer q=\"394349\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"394349\"]\r\n\r\nSolution\r\nThink about what we would have to square to get [latex]{x}^{2}[\/latex] . Algebraically, [latex]{\\left(?\\right)}^{2}={x}^{2}[\/latex]\r\n<table id=\"eip-id1168467419284\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\sqrt{{x}^{2}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Since [latex]{\\left(x\\right)}^{2}={x}^{2}[\/latex]<\/td>\r\n<td>[latex]x[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146637[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]\\sqrt{16{x}^{2}}[\/latex]\r\n[reveal-answer q=\"924940\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"924940\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168466092245\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\sqrt{16{x}^{2}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\text{Since}{\\left(4x\\right)}^{2}=16{x}^{2}[\/latex]<\/td>\r\n<td>[latex]4x[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146638[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]-\\sqrt{81{y}^{2}}[\/latex]\r\n[reveal-answer q=\"614317\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"614317\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168468389692\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-\\sqrt{81{y}^{2}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\text{Since}{\\left(9y\\right)}^{2}=81{y}^{2}[\/latex]<\/td>\r\n<td>[latex]-9y[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146639[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]\\sqrt{36{x}^{2}{y}^{2}}[\/latex]\r\n[reveal-answer q=\"164861\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"164861\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168466004603\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\sqrt{36{x}^{2}{y}^{2}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\text{Since}{\\left(6xy\\right)}^{2}=36{x}^{2}{y}^{2}[\/latex]<\/td>\r\n<td>[latex]6xy[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146640[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Simplify expressions with square roots using the order of operations<\/li>\n<li>Simplify expressions with square roots that contain variables<\/li>\n<\/ul>\n<\/div>\n<h3>Square Roots and the Order of Operations<\/h3>\n<p>When using the order of operations to simplify an expression that has square roots, we treat the radical sign as a grouping symbol. We simplify any expressions under the radical sign before performing other operations.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: (a)\u00a0 [latex]\\sqrt{25}+\\sqrt{144}[\/latex]\u00a0 \u00a0 (b) [latex]\\sqrt{25+144}[\/latex]<\/p>\n<p>Solution<\/p>\n<table id=\"eip-id1168466048150\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>(a) Use the order of operations.<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\sqrt{25}+\\sqrt{144}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify each radical.<\/td>\n<td>[latex]5+12[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add.<\/td>\n<td>[latex]17[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168469785439\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>(b) Use the order of operations.<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\sqrt{25+144}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add under the radical sign.<\/td>\n<td>[latex]\\sqrt{169}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]13[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146635\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146635&theme=oea&iframe_resize_id=ohm146635&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm146636\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146636&theme=oea&iframe_resize_id=ohm146636&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Notice the different answers in parts (a) and (b) of the example above. It is important to follow the order of operations correctly. In (a), we took each square root first and then added them. In (b), we added under the radical sign first and then found the square root.<\/p>\n<h2>Simplify Variable Expressions with Square Roots<\/h2>\n<p>Expressions with square root that we have looked at so far have not had any variables. What happens when we have to find a square root of a variable expression?<\/p>\n<p>Consider [latex]\\sqrt{9{x}^{2}}[\/latex], where [latex]x\\ge 0[\/latex]. Can you think of an expression whose square is [latex]9{x}^{2}?[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{ccc}\\hfill {\\left(?\\right)}^{2}& =& 9{x}^{2}\\hfill \\\\ \\hfill {\\left(3x\\right)}^{2}& =& 9{x}^{2}\\text{ so }\\sqrt{9{x}^{2}}=3x\\hfill \\end{array}[\/latex]<\/p>\n<p>When we use a variable in a square root expression, for our work, we will assume that the variable represents a non-negative number. In every example and exercise that follows, each variable in a square root expression is greater than or equal to zero.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]\\sqrt{{x}^{2}}[\/latex], where [latex]x\\ge 0[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q394349\">Show Solution<\/span><\/p>\n<div id=\"q394349\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<br \/>\nThink about what we would have to square to get [latex]{x}^{2}[\/latex] . Algebraically, [latex]{\\left(?\\right)}^{2}={x}^{2}[\/latex]<\/p>\n<table id=\"eip-id1168467419284\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]\\sqrt{{x}^{2}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Since [latex]{\\left(x\\right)}^{2}={x}^{2}[\/latex]<\/td>\n<td>[latex]x[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146637\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146637&theme=oea&iframe_resize_id=ohm146637&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: [latex]\\sqrt{16{x}^{2}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q924940\">Show Solution<\/span><\/p>\n<div id=\"q924940\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168466092245\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]\\sqrt{16{x}^{2}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\text{Since}{\\left(4x\\right)}^{2}=16{x}^{2}[\/latex]<\/td>\n<td>[latex]4x[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146638\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146638&theme=oea&iframe_resize_id=ohm146638&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: [latex]-\\sqrt{81{y}^{2}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q614317\">Show Solution<\/span><\/p>\n<div id=\"q614317\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168468389692\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]-\\sqrt{81{y}^{2}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\text{Since}{\\left(9y\\right)}^{2}=81{y}^{2}[\/latex]<\/td>\n<td>[latex]-9y[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146639\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146639&theme=oea&iframe_resize_id=ohm146639&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: [latex]\\sqrt{36{x}^{2}{y}^{2}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q164861\">Show Solution<\/span><\/p>\n<div id=\"q164861\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168466004603\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]\\sqrt{36{x}^{2}{y}^{2}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\text{Since}{\\left(6xy\\right)}^{2}=36{x}^{2}{y}^{2}[\/latex]<\/td>\n<td>[latex]6xy[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146640\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146640&theme=oea&iframe_resize_id=ohm146640&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-3644\">\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>Question ID 146640, 146639, 146638, 146637. <strong>Authored 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><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Prealgebra. <strong>Provided by<\/strong>: OpenStax. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t 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