{"id":4675,"date":"2023-06-19T17:10:40","date_gmt":"2023-06-19T17:10:40","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=4675"},"modified":"2024-10-18T20:50:58","modified_gmt":"2024-10-18T20:50:58","slug":"numbers-and-their-applications-background-youll-need-2","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/numbers-and-their-applications-background-youll-need-2\/","title":{"raw":"Numbers and Their Applications: Background You\u2019ll Need 2","rendered":"Numbers and Their Applications: Background You\u2019ll Need 2"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Use PEMDAS&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:12929,&quot;3&quot;:{&quot;1&quot;:0},&quot;10&quot;:0,&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;,&quot;16&quot;:10}\">Use PEMDAS<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Order of Operations<\/h2>\r\n<p>You may or may not recall the order of operations for applying several mathematical operations to one expression. Just as it is a social convention for us to drive on the right-hand side of the road, the order of operations is a set of conventions used to provide order when you are required to use several mathematical operations for one expression.<\/p>\r\n<section class=\"textbox questionHelp\">\r\n<p><strong>How to: Perform the Order of Operations<\/strong><\/p>\r\n<ol>\r\n\t<li>Perform all operations within grouping symbols first. Grouping symbols include parentheses ( ), brackets [ ], braces { }, and fraction bars.<\/li>\r\n\t<li>Evaluate exponents or square roots.<\/li>\r\n\t<li>Multiply or divide, from left to right.<\/li>\r\n\t<li>Add or subtract, from left to right.<\/li>\r\n<\/ol>\r\n<p>This order of operations is true for all real numbers.<\/p>\r\n<\/section>\r\n<section class=\"textbox proTip\">Some people use a saying to help them remember the order of operations. This saying is called PEMDAS or <strong>P<\/strong>lease <strong>E<\/strong>xcuse <strong>M<\/strong>y <strong>D<\/strong>ear <strong>A<\/strong>unt <strong>S<\/strong>ally. The first letter of each word begins with the same letter of an arithmetic operation. <br \/>\r\n<br \/>\r\n<ul>\r\n\t<li><strong>P<\/strong>lease [latex] \\displaystyle \\Rightarrow [\/latex] <strong>P<\/strong>arentheses (and other grouping symbols)<\/li>\r\n\t<li><strong>E<\/strong>xcuse [latex] \\displaystyle \\Rightarrow [\/latex] <strong>E<\/strong>xponents<\/li>\r\n\t<li><strong>M<\/strong>y <strong>D<\/strong>ear [latex] \\displaystyle \\Rightarrow [\/latex] <strong>M<\/strong>ultiplication and <strong>D<\/strong>ivision (from left to right)<\/li>\r\n\t<li><strong>A<\/strong>unt <strong>S<\/strong>ally [latex] \\displaystyle \\Rightarrow [\/latex] <strong>A<\/strong>ddition and <strong>S<\/strong>ubtraction (from left to right)<\/li>\r\n<\/ul>\r\n\r\nEven though multiplication comes before division in the saying, division could be performed first. Which is performed first, between multiplication and division, is determined by which comes first when reading from left to right. The same is true of addition and subtraction. Don't let the saying confuse you about this!<\/section>\r\n<section class=\"textbox example\">Simplify the following:<center>[latex]7\u20135+3\\cdot8[\/latex]<\/center>[reveal-answer q=\"796973\"]Show Answer[\/reveal-answer] [hidden-answer a=\"796973\"]According to the order of operations, multiplication comes before addition and subtraction.<br \/>\r\n<br \/>\r\nMultiply [latex]3\\cdot8[\/latex].\r\n\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}7\u20135+3\\cdot8\\\\7\u20135+24\\end{array}[\/latex]<\/p>\r\n\r\nNow, add and subtract from left to right. [latex]7\u20135[\/latex] comes first.\r\n\r\n<p style=\"text-align: center;\">[latex]2+24[\/latex]<\/p>\r\n<p>Finally, add.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}2+24=26\\\\7\u20135+3\\cdot8=26\\end{array}[\/latex]<\/p>\r\n\r\n[\/hidden-answer]<\/section>\r\n<h2>Exponents and Square Roots<\/h2>\r\n<p>In this section, we expand our skills with applying the order of operation rules to expressions with\u00a0exponents and square roots.\u00a0If the expression has exponents or square roots, they are to be performed a<i>fter <\/i>parentheses and other grouping symbols have been simplified and <i>before <\/i>any multiplication, division, subtraction, and addition that are outside the parentheses or other grouping symbols.<\/p>\r\n<section class=\"textbox recall\">An expression such as [latex]7^{2}[\/latex] is <strong>exponential notation<\/strong> for [latex]7\\cdot7[\/latex]. <br \/>\r\n<br \/>\r\nExponential notation has two parts: the <strong>base<\/strong> and the <strong>exponent<\/strong> or the <strong>power<\/strong>. In [latex]7^{2}[\/latex], [latex]7[\/latex]is the base and [latex]2[\/latex] is the exponent; the exponent determines how many times the base is multiplied by itself.<br \/>\r\n<br \/>\r\nExponents are a way to represent repeated multiplication; the order of operations places it <i>before <\/i>any other multiplication, division, subtraction, and addition is performed.<\/section>\r\n<section class=\"textbox example\">Simplify the following:<center>[latex]3^{2}\\cdot2^{3}[\/latex]<\/center>[reveal-answer q=\"968408\"]Show Answer[\/reveal-answer] [hidden-answer a=\"968408\"]This problem has exponents and multiplication in it. According to the order of operations, simplifying\u00a0[latex]3^{2}[\/latex]\u00a0and [latex]2^{3}[\/latex]\u00a0comes before multiplication.\r\n\r\n<p style=\"text-align: center;\">[latex]3^{2}\\cdot2^{3}[\/latex]<\/p>\r\n\r\n[latex] {{3}^{2}}[\/latex] is [latex]3\\cdot3[\/latex], which equals\u00a0[latex]9[\/latex].\r\n\r\n<p style=\"text-align: center;\">[latex] 9\\cdot {{2}^{3}}[\/latex]<\/p>\r\n\r\n[latex] {{2}^{3}}[\/latex] is [latex]2\\cdot2\\cdot2[\/latex], which equals\u00a0[latex]8[\/latex].\r\n\r\n<p style=\"text-align: center;\">[latex] 9\\cdot 8[\/latex]<\/p>\r\n\r\nMultiply.\r\n\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c} 9\\cdot 8=72\\\\{{3}^{2}}\\cdot {{2}^{3}}=72 \\end{array}[\/latex]<\/p>\r\n\r\n[\/hidden-answer]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]8373[\/ohm2_question]<\/section>\r\n<p>In the next example we will simplify an expression that has a square root.<\/p>\r\n<section class=\"textbox example\">Simplify the following:<center>\u00a0[latex]\\Large\\frac{\\sqrt{7+2}+2^2}{(8)(4)-11}[\/latex]<\/center>[reveal-answer q=\"484589\"]Show Answer[\/reveal-answer] [hidden-answer a=\"484589\"]This problem has all the operations to consider with the order of operations.Grouping symbols are handled first, in this case the fraction bar. We will simplify the top and bottom separately. To simplify the top:\r\n\r\n<p style=\"text-align: center;\">[latex]\\sqrt{7+2}+2^2[\/latex]<\/p>\r\n\r\nAdd the numbers inside the square root, simplify the result and [latex]2^2[\/latex].\r\n\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\sqrt{7+2}+2^2\\\\\\\\=\\sqrt{9}+4\\\\\\\\=3+4=7\\end{array}[\/latex]<\/p>\r\n\r\nTo simplify the bottom:\r\n\r\n<p style=\"text-align: center;\">[latex](8)(4)-11[\/latex]<\/p>\r\n\r\nMultiply [latex]8[\/latex] and [latex]4[\/latex] first, then subtract [latex]11[\/latex].\r\n\r\n<p style=\"text-align: center;\">[latex](8)(4)-11=32-11=21[\/latex]<\/p>\r\n\r\nNow put the fraction back together to see if any more simplifying needs to be done. [latex]\\Large\\frac{7}{21}[\/latex], this can be reduced to [latex]\\Large\\frac{1}{3}[\/latex] .\r\n\r\n<p style=\"text-align: center;\">[latex]\\Large\\frac{\\sqrt{7+2}+2^2}{(8)(4)-11}=\\frac{1}{3}[\/latex]<\/p>\r\n\r\n[\/hidden-answer]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]8389[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Use PEMDAS&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:12929,&quot;3&quot;:{&quot;1&quot;:0},&quot;10&quot;:0,&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;,&quot;16&quot;:10}\">Use PEMDAS<\/span><\/li>\n<\/ul>\n<\/section>\n<h2>Order of Operations<\/h2>\n<p>You may or may not recall the order of operations for applying several mathematical operations to one expression. Just as it is a social convention for us to drive on the right-hand side of the road, the order of operations is a set of conventions used to provide order when you are required to use several mathematical operations for one expression.<\/p>\n<section class=\"textbox questionHelp\">\n<p><strong>How to: Perform the Order of Operations<\/strong><\/p>\n<ol>\n<li>Perform all operations within grouping symbols first. Grouping symbols include parentheses ( ), brackets [ ], braces { }, and fraction bars.<\/li>\n<li>Evaluate exponents or square roots.<\/li>\n<li>Multiply or divide, from left to right.<\/li>\n<li>Add or subtract, from left to right.<\/li>\n<\/ol>\n<p>This order of operations is true for all real numbers.<\/p>\n<\/section>\n<section class=\"textbox proTip\">Some people use a saying to help them remember the order of operations. This saying is called PEMDAS or <strong>P<\/strong>lease <strong>E<\/strong>xcuse <strong>M<\/strong>y <strong>D<\/strong>ear <strong>A<\/strong>unt <strong>S<\/strong>ally. The first letter of each word begins with the same letter of an arithmetic operation. <\/p>\n<ul>\n<li><strong>P<\/strong>lease [latex]\\displaystyle \\Rightarrow[\/latex] <strong>P<\/strong>arentheses (and other grouping symbols)<\/li>\n<li><strong>E<\/strong>xcuse [latex]\\displaystyle \\Rightarrow[\/latex] <strong>E<\/strong>xponents<\/li>\n<li><strong>M<\/strong>y <strong>D<\/strong>ear [latex]\\displaystyle \\Rightarrow[\/latex] <strong>M<\/strong>ultiplication and <strong>D<\/strong>ivision (from left to right)<\/li>\n<li><strong>A<\/strong>unt <strong>S<\/strong>ally [latex]\\displaystyle \\Rightarrow[\/latex] <strong>A<\/strong>ddition and <strong>S<\/strong>ubtraction (from left to right)<\/li>\n<\/ul>\n<p>Even though multiplication comes before division in the saying, division could be performed first. Which is performed first, between multiplication and division, is determined by which comes first when reading from left to right. The same is true of addition and subtraction. Don&#8217;t let the saying confuse you about this!<\/section>\n<section class=\"textbox example\">Simplify the following:<\/p>\n<div style=\"text-align: center;\">[latex]7\u20135+3\\cdot8[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q796973\">Show Answer<\/button> <\/p>\n<div id=\"q796973\" class=\"hidden-answer\" style=\"display: none\">According to the order of operations, multiplication comes before addition and subtraction.<\/p>\n<p>Multiply [latex]3\\cdot8[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}7\u20135+3\\cdot8\\\\7\u20135+24\\end{array}[\/latex]<\/p>\n<p>Now, add and subtract from left to right. [latex]7\u20135[\/latex] comes first.<\/p>\n<p style=\"text-align: center;\">[latex]2+24[\/latex]<\/p>\n<p>Finally, add.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}2+24=26\\\\7\u20135+3\\cdot8=26\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h2>Exponents and Square Roots<\/h2>\n<p>In this section, we expand our skills with applying the order of operation rules to expressions with\u00a0exponents and square roots.\u00a0If the expression has exponents or square roots, they are to be performed a<i>fter <\/i>parentheses and other grouping symbols have been simplified and <i>before <\/i>any multiplication, division, subtraction, and addition that are outside the parentheses or other grouping symbols.<\/p>\n<section class=\"textbox recall\">An expression such as [latex]7^{2}[\/latex] is <strong>exponential notation<\/strong> for [latex]7\\cdot7[\/latex]. <\/p>\n<p>Exponential notation has two parts: the <strong>base<\/strong> and the <strong>exponent<\/strong> or the <strong>power<\/strong>. In [latex]7^{2}[\/latex], [latex]7[\/latex]is the base and [latex]2[\/latex] is the exponent; the exponent determines how many times the base is multiplied by itself.<\/p>\n<p>Exponents are a way to represent repeated multiplication; the order of operations places it <i>before <\/i>any other multiplication, division, subtraction, and addition is performed.<\/section>\n<section class=\"textbox example\">Simplify the following:<\/p>\n<div style=\"text-align: center;\">[latex]3^{2}\\cdot2^{3}[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q968408\">Show Answer<\/button> <\/p>\n<div id=\"q968408\" class=\"hidden-answer\" style=\"display: none\">This problem has exponents and multiplication in it. According to the order of operations, simplifying\u00a0[latex]3^{2}[\/latex]\u00a0and [latex]2^{3}[\/latex]\u00a0comes before multiplication.<\/p>\n<p style=\"text-align: center;\">[latex]3^{2}\\cdot2^{3}[\/latex]<\/p>\n<p>[latex]{{3}^{2}}[\/latex] is [latex]3\\cdot3[\/latex], which equals\u00a0[latex]9[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]9\\cdot {{2}^{3}}[\/latex]<\/p>\n<p>[latex]{{2}^{3}}[\/latex] is [latex]2\\cdot2\\cdot2[\/latex], which equals\u00a0[latex]8[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]9\\cdot 8[\/latex]<\/p>\n<p>Multiply.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c} 9\\cdot 8=72\\\\{{3}^{2}}\\cdot {{2}^{3}}=72 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm8373\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=8373&theme=lumen&iframe_resize_id=ohm8373&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p>In the next example we will simplify an expression that has a square root.<\/p>\n<section class=\"textbox example\">Simplify the following:<\/p>\n<div style=\"text-align: center;\">\u00a0[latex]\\Large\\frac{\\sqrt{7+2}+2^2}{(8)(4)-11}[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q484589\">Show Answer<\/button> <\/p>\n<div id=\"q484589\" class=\"hidden-answer\" style=\"display: none\">This problem has all the operations to consider with the order of operations.Grouping symbols are handled first, in this case the fraction bar. We will simplify the top and bottom separately. To simplify the top:<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt{7+2}+2^2[\/latex]<\/p>\n<p>Add the numbers inside the square root, simplify the result and [latex]2^2[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\sqrt{7+2}+2^2\\\\\\\\=\\sqrt{9}+4\\\\\\\\=3+4=7\\end{array}[\/latex]<\/p>\n<p>To simplify the bottom:<\/p>\n<p style=\"text-align: center;\">[latex](8)(4)-11[\/latex]<\/p>\n<p>Multiply [latex]8[\/latex] and [latex]4[\/latex] first, then subtract [latex]11[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex](8)(4)-11=32-11=21[\/latex]<\/p>\n<p>Now put the fraction back together to see if any more simplifying needs to be done. [latex]\\Large\\frac{7}{21}[\/latex], this can be reduced to [latex]\\Large\\frac{1}{3}[\/latex] .<\/p>\n<p style=\"text-align: center;\">[latex]\\Large\\frac{\\sqrt{7+2}+2^2}{(8)(4)-11}=\\frac{1}{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm8389\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=8389&theme=lumen&iframe_resize_id=ohm8389&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":16,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Unit 9: Real Numbers, 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\":\"\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":54,"module-header":"background_you_need","content_attributions":[{"type":"original","description":"Revision and Adaptation","author":"","organization":"Lumen Learning","url":"","project":"","license":"cc-by","license_terms":""},{"type":"cc","description":"Unit 9: Real Numbers, 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":""}],"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\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/4675"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/users\/16"}],"version-history":[{"count":39,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/4675\/revisions"}],"predecessor-version":[{"id":15186,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/4675\/revisions\/15186"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/parts\/54"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/4675\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=4675"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=4675"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=4675"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=4675"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}