{"id":3642,"date":"2020-01-28T01:35:59","date_gmt":"2020-01-28T01:35:59","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/mathforlibscoreq\/?post_type=chapter&#038;p=3642"},"modified":"2020-02-03T10:53:01","modified_gmt":"2020-02-03T10:53:01","slug":"estimating-and-approximating-square-roots","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/mathforliberalartscorequisite\/chapter\/estimating-and-approximating-square-roots\/","title":{"raw":"Estimating and Approximating Square Roots","rendered":"Estimating and Approximating Square Roots"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Estimate a square root that is not a perfect square<\/li>\r\n \t<li>Approximate square roots with a calculator<\/li>\r\n<\/ul>\r\n<\/div>\r\n<p>So far we have only worked with square roots of perfect squares. The square roots of other numbers are not whole numbers.<\/p>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221841\/CNX_BMath_Figure_05_07_007_img.png\" alt=\"A table is shown with 2 columns. The first column is labeled \" \/>\r\nWe might conclude that the square roots of numbers between [latex]4[\/latex] and [latex]9[\/latex] will be between [latex]2[\/latex] and [latex]3[\/latex], and they will not be whole numbers. Based on the pattern in the table above, we could say that [latex]\\sqrt{5}[\/latex] is between [latex]2[\/latex] and [latex]3[\/latex]. Using inequality symbols, we write\r\n<p style=\"text-align: center\">[latex]2&lt;\\sqrt{5}&lt;3[\/latex]<\/p>\r\n\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nEstimate [latex]\\sqrt{60}[\/latex] between two consecutive whole numbers.\r\n\r\nSolution\r\nThink of the perfect squares closest to [latex]60[\/latex]. Make a small table of these perfect squares and their squares roots.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221845\/CNX_BMath_Figure_05_07_006_img.png\" alt=\"A table is shown with 2 columns. The first column is labeled \" \/>\r\n<table id=\"eip-id1168467173482\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>[latex]\\text{Locate 60 between two consecutive perfect squares.}[\/latex]<\/td>\r\n<td>[latex]49&lt;60&lt;64[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\sqrt{60}\\text{ is between their square roots.}[\/latex]<\/td>\r\n<td>[latex]7&lt;\\sqrt{60}&lt;8[\/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]146633[\/ohm_question]\r\n\r\n<\/div>\r\nIn the next video you will see more examples of how to estimate a square root between two consecutive whole numbers.\r\n\r\nhttps:\/\/youtu.be\/-ViX7ZtXP8E\r\n<h3>Approximate Square Roots with a Calculator<\/h3>\r\nThere are mathematical methods to approximate square roots, but it is much more convenient to use a calculator to find square roots. Find the [latex]\\sqrt{\\phantom{0}}[\/latex] or [latex]\\sqrt{x}[\/latex] key on your calculator. You will to use this key to approximate square roots. When you use your calculator to find the square root of a number that is not a perfect square, the answer that you see is not the exact number. It is an approximation, to the number of digits shown on your calculator\u2019s display. The symbol for an approximation is [latex]\\approx [\/latex] and it is read <em>approximately<\/em>.\r\n\r\nSuppose your calculator has a [latex]\\text{10-digit}[\/latex] display. Using it to find the square root of [latex]5[\/latex] will give [latex]2.236067977[\/latex]. This is the approximate square root of [latex]5[\/latex]. When we report the answer, we should use the \"approximately equal to\" sign instead of an equal sign.\r\n<p style=\"text-align: center\">[latex]\\sqrt{5}\\approx 2.236067978[\/latex]<\/p>\r\nYou will seldom use this many digits for applications in algebra. So, if you wanted to round [latex]\\sqrt{5}[\/latex] to two decimal places, you would write\r\n<p style=\"text-align: center\">[latex]\\sqrt{5}\\approx 2.24[\/latex]<\/p>\r\nHow do we know these values are approximations and not the exact values? Look at what happens when we square them.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{ccc}\\hfill {2.236067978}^{2}&amp; =&amp; 5.000000002\\hfill \\\\ \\hfill {2.24}^{2}&amp; =&amp; 5.0176\\hfill \\end{array}[\/latex]<\/p>\r\nThe squares are close, but not exactly equal, to [latex]5[\/latex].\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nRound [latex]\\sqrt{17}[\/latex] to two decimal places using a calculator.\r\n[reveal-answer q=\"155113\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"155113\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168468614270\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>[latex]\\sqrt{17}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the calculator square root key.<\/td>\r\n<td>[latex]4.123105626[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Round to two decimal places.<\/td>\r\n<td>[latex]4.12[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\sqrt{17}\\approx 4.12[\/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]146634[\/ohm_question]\r\n\r\n<\/div>\r\nIn the next video you will see more examples of how to use a calculator to estimate the square root of a number.\r\n\r\nhttps:\/\/youtu.be\/eHOrbHt6AD4","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Estimate a square root that is not a perfect square<\/li>\n<li>Approximate square roots with a calculator<\/li>\n<\/ul>\n<\/div>\n<p>So far we have only worked with square roots of perfect squares. The square roots of other numbers are not whole numbers.<\/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\/24221841\/CNX_BMath_Figure_05_07_007_img.png\" alt=\"A table is shown with 2 columns. The first column is labeled\" \/><br \/>\nWe might conclude that the square roots of numbers between [latex]4[\/latex] and [latex]9[\/latex] will be between [latex]2[\/latex] and [latex]3[\/latex], and they will not be whole numbers. Based on the pattern in the table above, we could say that [latex]\\sqrt{5}[\/latex] is between [latex]2[\/latex] and [latex]3[\/latex]. Using inequality symbols, we write<\/p>\n<p style=\"text-align: center\">[latex]2<\\sqrt{5}<3[\/latex]<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Estimate [latex]\\sqrt{60}[\/latex] between two consecutive whole numbers.<\/p>\n<p>Solution<br \/>\nThink of the perfect squares closest to [latex]60[\/latex]. Make a small table of these perfect squares and their squares roots.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221845\/CNX_BMath_Figure_05_07_006_img.png\" alt=\"A table is shown with 2 columns. The first column is labeled\" \/><\/p>\n<table id=\"eip-id1168467173482\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>[latex]\\text{Locate 60 between two consecutive perfect squares.}[\/latex]<\/td>\n<td>[latex]49<60<64[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\sqrt{60}\\text{ is between their square roots.}[\/latex]<\/td>\n<td>[latex]7<\\sqrt{60}<8[\/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=\"ohm146633\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146633&theme=oea&iframe_resize_id=ohm146633&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the next video you will see more examples of how to estimate a square root between two consecutive whole numbers.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Determine Consecutive Whole Numbers  A Square Root is Between\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/-ViX7ZtXP8E?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h3>Approximate Square Roots with a Calculator<\/h3>\n<p>There are mathematical methods to approximate square roots, but it is much more convenient to use a calculator to find square roots. Find the [latex]\\sqrt{\\phantom{0}}[\/latex] or [latex]\\sqrt{x}[\/latex] key on your calculator. You will to use this key to approximate square roots. When you use your calculator to find the square root of a number that is not a perfect square, the answer that you see is not the exact number. It is an approximation, to the number of digits shown on your calculator\u2019s display. The symbol for an approximation is [latex]\\approx[\/latex] and it is read <em>approximately<\/em>.<\/p>\n<p>Suppose your calculator has a [latex]\\text{10-digit}[\/latex] display. Using it to find the square root of [latex]5[\/latex] will give [latex]2.236067977[\/latex]. This is the approximate square root of [latex]5[\/latex]. When we report the answer, we should use the &#8220;approximately equal to&#8221; sign instead of an equal sign.<\/p>\n<p style=\"text-align: center\">[latex]\\sqrt{5}\\approx 2.236067978[\/latex]<\/p>\n<p>You will seldom use this many digits for applications in algebra. So, if you wanted to round [latex]\\sqrt{5}[\/latex] to two decimal places, you would write<\/p>\n<p style=\"text-align: center\">[latex]\\sqrt{5}\\approx 2.24[\/latex]<\/p>\n<p>How do we know these values are approximations and not the exact values? Look at what happens when we square them.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{ccc}\\hfill {2.236067978}^{2}& =& 5.000000002\\hfill \\\\ \\hfill {2.24}^{2}& =& 5.0176\\hfill \\end{array}[\/latex]<\/p>\n<p>The squares are close, but not exactly equal, to [latex]5[\/latex].<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Round [latex]\\sqrt{17}[\/latex] to two decimal places using a calculator.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q155113\">Show Solution<\/span><\/p>\n<div id=\"q155113\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168468614270\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>[latex]\\sqrt{17}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the calculator square root key.<\/td>\n<td>[latex]4.123105626[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Round to two decimal places.<\/td>\n<td>[latex]4.12[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\sqrt{17}\\approx 4.12[\/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=\"ohm146634\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146634&theme=oea&iframe_resize_id=ohm146634&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the next video you will see more examples of how to use a calculator to estimate the square root of a number.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex:  Estimating Square Roots with a Calculator\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/eHOrbHt6AD4?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/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-3642\">\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: 146634, 146633, 146620. <strong>Authored by<\/strong>: Alyson Day. <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>: IMathAS Community License CC-BY + GPL<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Determine Consecutive Whole Numbers A Square Root is Between. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/-ViX7ZtXP8E\">https:\/\/youtu.be\/-ViX7ZtXP8E<\/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>Ex: Estimating Square Roots with a Calculator. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/eHOrbHt6AD4\">https:\/\/youtu.be\/eHOrbHt6AD4<\/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, 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 <\/section>","protected":false},"author":25777,"menu_order":7,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Prealgebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757\"},{\"type\":\"cc\",\"description\":\"Determine Consecutive Whole Numbers A Square Root is Between\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/-ViX7ZtXP8E\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex: Estimating Square Roots with a Calculator\",\"author\":\"James Sousa 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GPL\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-3642","chapter","type-chapter","status-publish","hentry"],"part":50,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/3642","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/mathforliberalartscorequisite\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/mathforliberalartscorequisite\/wp-json\/wp\/v2\/users\/25777"}],"version-history":[{"count":2,"href":"https:\/\/courses.lumenlearning.com\/mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/3642\/revisions"}],"predecessor-version":[{"id":3713,"href":"https:\/\/courses.lumenlearning.com\/mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/3642\/revisions\/3713"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/parts\/50"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/3642\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/mathforliberalartscorequisite\/wp-json\/wp\/v2\/media?parent=3642"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=3642"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/mathforliberalartscorequisite\/wp-json\/wp\/v2\/contributor?post=3642"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/mathforliberalartscorequisite\/wp-json\/wp\/v2\/license?post=3642"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}