{"id":505,"date":"2024-04-22T19:13:16","date_gmt":"2024-04-22T19:13:16","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=505"},"modified":"2024-11-20T00:51:54","modified_gmt":"2024-11-20T00:51:54","slug":"introduction-to-real-numbers-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/introduction-to-real-numbers-learn-it-2\/","title":{"raw":"Introduction to Real Numbers: Learn It 2","rendered":"Introduction to Real Numbers: Learn It 2"},"content":{"raw":"<h2>Order of Operations<\/h2>\r\nTo evaluate a mathematical expression, we perform the various operations. However, we do not perform them in any random order. We use the order of operations. This is a sequence of rules for evaluating such expressions.\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>order of operations<\/h3>\r\nThe <strong>order of operations<\/strong> is a set of rules used in mathematics to determine the sequence in which operations should be performed to correctly solve an expression.\r\n\r\n&nbsp;\r\n\r\nThe standard order in which these operations must be carried out is often remembered by the acronym <strong>PEMDAS<\/strong>:\r\n<ol>\r\n \t<li><strong>P(<\/strong>arentheses): First, perform all operations inside parentheses ( ) or other grouping symbols like brackets [ ] and braces { }. Additionally, fraction bars, radicals, and absolute value bars are treated as grouping symbols.<\/li>\r\n \t<li><strong>E(<\/strong>xponents): Next, solve any [pb_glossary id=\"506\"]exponents[\/pb_glossary] ([latex]a^n = a \\cdot a \\cdot ... \\cdot a[\/latex]) and roots.<\/li>\r\n \t<li><strong>M(<\/strong>ultiplication) and <strong>D(<\/strong>ivision): Then, perform all multiplication and division from left to right, as they appear in the expression. These operations are of equal precedence and are carried out in the order they occur from left to right.<\/li>\r\n \t<li><strong>A(<\/strong>ddition) and <strong>S(<\/strong>ubtraction): Lastly, perform all addition and subtraction from left to right, as they appear. Like multiplication and division, these operations are of equal precedence and should be carried out from left to right.<\/li>\r\n<\/ol>\r\nThis order of operations is true for all real numbers.\r\n\r\n<\/section><section class=\"textbox questionHelp\" aria-label=\"Question Help\"><b>How to: <strong>Given a mathematical expression, simplify it using the order of operations.<\/strong><\/b>\r\n<ol id=\"fs-id1374893\" class=\"os-stepwise\" type=\"1\">\r\n \t<li><span class=\"os-stepwise-content\">Simplify any expressions within grouping symbols.<\/span><\/li>\r\n \t<li><span class=\"os-stepwise-content\">Simplify any expressions containing exponents or radicals.<\/span><\/li>\r\n \t<li><span class=\"os-stepwise-content\">Perform any multiplication and division in order, from left to right.<\/span><\/li>\r\n \t<li><span class=\"os-stepwise-content\">Perform any addition and subtraction in order, from left to right.<\/span><\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox example\">Simplify the expression [latex]8+(3\\cdot2^2)-3[\/latex].[reveal-answer q=\"575668\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"575668\"]\r\n<ul>\r\n \t<li><strong>Parentheses<\/strong>: First, solve the expression inside the parentheses.\r\n<ul>\r\n \t<li>Inside the parentheses: [latex]3\\cdot 2^2[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><strong>Exponents<\/strong>: Solve any exponents.\r\n<ul>\r\n \t<li>[latex]2^2 = 2 \\cdot 2 = 4[\/latex], so the expression inside the parentheses becomes [latex]3\\cdot 4[\/latex].<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><strong>Multiplication<\/strong>: Then, perform the multiplication.\r\n<ul>\r\n \t<li>[latex]3\\cdot4 =12[\/latex], so now the expression is [latex]8+12-3[\/latex].<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><strong>Addition and Subtraction<\/strong>: Finally, perform the addition and subtraction from left to right.\r\n<ul>\r\n \t<li><span class=\"math math-inline\"><span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">[latex]8+12-3[\/latex]<\/span><\/span><\/span><\/span><\/span><\/li>\r\n \t<li><span class=\"math math-inline\"><span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">[latex]20-3[\/latex]<\/span><\/span><\/span><\/span><\/span><\/li>\r\n \t<li><span class=\"math math-inline\"><span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">[latex]17[\/latex]<\/span><\/span><\/span><\/span><\/span><\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\nTherefore, [latex]8+(3\\cdot2^2)-3 = 17[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>For some complicated expressions, several passes through the order of operations will be needed. For instance, there may be a radical expression inside parentheses that must be simplified before the parentheses are evaluated. Following the order of operations ensures that anyone simplifying the same mathematical expression will get the same result.\r\n\r\n<section class=\"textbox example\">Use the order of operations to evaluate each of the following expressions.\r\n<ol>\r\n \t<li>[latex]{\\left(3\\cdot 2\\right)}^{2}-4\\left(6+2\\right)[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{{5}^{2}-4}{7}-\\sqrt{11 - 2}[\/latex]<\/li>\r\n \t<li>[latex]6-|5 - 8|+3\\left(4 - 1\\right)[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{14 - 3\\cdot 2}{2\\cdot 5-{3}^{2}}[\/latex]<\/li>\r\n \t<li>[latex]7\\left(5\\cdot 3\\right)-2\\left[\\left(6 - 3\\right)-{4}^{2}\\right]+1[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"371324\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"371324\"]\r\n\r\n1.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\left(3\\cdot 2\\right)^{2} &amp; =\\left(6\\right)^{2}-4\\left(8\\right) &amp;&amp; \\text{Simplify parentheses} \\\\ &amp; =36-4\\left(8\\right) &amp;&amp; \\text{Simplify exponent} \\\\ &amp; =36-32 &amp;&amp; \\text{Simplify multiplication} \\\\ &amp; =4 &amp;&amp; \\text{Simplify subtraction}\\end{align}[\/latex]<\/p>\r\n2.\r\n<p class=\"p1\" style=\"text-align: center;\"><span class=\"s1\">[latex]\\begin{align}\\frac{5^{2}-4}{7}-\\sqrt{11-2} &amp; =\\frac{5^{2}-4}{7}-\\sqrt{9} &amp;&amp; \\text{Simplify grouping systems (radical)} \\\\ &amp; =\\frac{5^{2}-4}{7}-3 &amp;&amp; \\text{Simplify radical} \\\\ &amp; =\\frac{25-4}{7}-3 &amp;&amp; \\text{Simplify exponent} \\\\ &amp; =\\frac{21}{7}-3 &amp;&amp; \\text{Simplify subtraction in numerator} \\\\ &amp; =3-3 &amp;&amp; \\text{Simplify division} \\\\ &amp; =0 &amp;&amp; \\text{Simplify subtraction}\\end{align}[\/latex]<\/span><\/p>\r\nNote that in the first step, the radical is treated as a grouping symbol, like parentheses. Also, in the third step, the fraction bar is considered a grouping symbol so the numerator is considered to be grouped.\r\n\r\n3.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}6-|5-8|+3\\left(4-1\\right) &amp; =6-|-3|+3\\left(3\\right) &amp;&amp; \\text{Simplify inside grouping system} \\\\ &amp; =6-3+3\\left(3\\right) &amp;&amp; \\text{Simplify absolute value} \\\\ &amp; =6-3+9 &amp;&amp; \\text{Simplify multiplication} \\\\ &amp; =3+9 &amp;&amp; \\text{Simplify subtraction} \\\\ &amp; =12 &amp;&amp; \\text{Simplify addition}\\end{align}[\/latex]<\/p>\r\n4.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\frac{14-3\\cdot2}{2\\cdot5-3^{2}} &amp; =\\frac{14-3\\cdot2}{2\\cdot5-9} &amp;&amp; \\text{Simplify exponent} \\\\ &amp; =\\frac{14-6}{10-9} &amp;&amp; \\text{Simplify products} \\\\ &amp; =\\frac{8}{1} &amp;&amp; \\text{Simplify quotient} \\\\ &amp; =8 &amp;&amp; \\text{Simplify quotient}\\end{align}[\/latex]\r\nIn this example, the fraction bar separates the numerator and denominator, which we simplify separately until the last step.<\/p>\r\n5.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}7\\left(5\\cdot3\\right)-2[\\left(6-3\\right)-4^{2}]+1 &amp; =7\\left(15\\right)-2[\\left(3\\right)-4^{2}]+1 &amp;&amp; \\text{Simplify inside parentheses} \\\\ &amp; 7\\left(15\\right)-2\\left(3-16\\right)+1 &amp;&amp; \\text{Simplify exponent} \\\\ &amp; =7\\left(15\\right)-2\\left(-13\\right)+1 &amp;&amp; \\text{Subtract} \\\\ &amp; =105+26+1 &amp;&amp; \\text{Multiply} \\\\ &amp; =132 &amp;&amp; \\text{Add}\\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]18712[\/ohm2_question]<\/section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]18713[\/ohm2_question]<\/section><section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]18734[\/ohm2_question]<\/section><\/section>","rendered":"<h2>Order of Operations<\/h2>\n<p>To evaluate a mathematical expression, we perform the various operations. However, we do not perform them in any random order. We use the order of operations. This is a sequence of rules for evaluating such expressions.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>order of operations<\/h3>\n<p>The <strong>order of operations<\/strong> is a set of rules used in mathematics to determine the sequence in which operations should be performed to correctly solve an expression.<\/p>\n<p>&nbsp;<\/p>\n<p>The standard order in which these operations must be carried out is often remembered by the acronym <strong>PEMDAS<\/strong>:<\/p>\n<ol>\n<li><strong>P(<\/strong>arentheses): First, perform all operations inside parentheses ( ) or other grouping symbols like brackets [ ] and braces { }. Additionally, fraction bars, radicals, and absolute value bars are treated as grouping symbols.<\/li>\n<li><strong>E(<\/strong>xponents): Next, solve any <a class=\"glossary-term\" aria-haspopup=\"dialog\" aria-describedby=\"definition\" href=\"#term_505_506\">exponents<\/a> ([latex]a^n = a \\cdot a \\cdot ... \\cdot a[\/latex]) and roots.<\/li>\n<li><strong>M(<\/strong>ultiplication) and <strong>D(<\/strong>ivision): Then, perform all multiplication and division from left to right, as they appear in the expression. These operations are of equal precedence and are carried out in the order they occur from left to right.<\/li>\n<li><strong>A(<\/strong>ddition) and <strong>S(<\/strong>ubtraction): Lastly, perform all addition and subtraction from left to right, as they appear. Like multiplication and division, these operations are of equal precedence and should be carried out 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 questionHelp\" aria-label=\"Question Help\"><b>How to: <strong>Given a mathematical expression, simplify it using the order of operations.<\/strong><\/b><\/p>\n<ol id=\"fs-id1374893\" class=\"os-stepwise\" type=\"1\">\n<li><span class=\"os-stepwise-content\">Simplify any expressions within grouping symbols.<\/span><\/li>\n<li><span class=\"os-stepwise-content\">Simplify any expressions containing exponents or radicals.<\/span><\/li>\n<li><span class=\"os-stepwise-content\">Perform any multiplication and division in order, from left to right.<\/span><\/li>\n<li><span class=\"os-stepwise-content\">Perform any addition and subtraction in order, from left to right.<\/span><\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\">Simplify the expression [latex]8+(3\\cdot2^2)-3[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q575668\">Show Answer<\/button><\/p>\n<div id=\"q575668\" class=\"hidden-answer\" style=\"display: none\">\n<ul>\n<li><strong>Parentheses<\/strong>: First, solve the expression inside the parentheses.\n<ul>\n<li>Inside the parentheses: [latex]3\\cdot 2^2[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li><strong>Exponents<\/strong>: Solve any exponents.\n<ul>\n<li>[latex]2^2 = 2 \\cdot 2 = 4[\/latex], so the expression inside the parentheses becomes [latex]3\\cdot 4[\/latex].<\/li>\n<\/ul>\n<\/li>\n<li><strong>Multiplication<\/strong>: Then, perform the multiplication.\n<ul>\n<li>[latex]3\\cdot4 =12[\/latex], so now the expression is [latex]8+12-3[\/latex].<\/li>\n<\/ul>\n<\/li>\n<li><strong>Addition and Subtraction<\/strong>: Finally, perform the addition and subtraction from left to right.\n<ul>\n<li><span class=\"math math-inline\"><span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">[latex]8+12-3[\/latex]<\/span><\/span><\/span><\/span><\/span><\/li>\n<li><span class=\"math math-inline\"><span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">[latex]20-3[\/latex]<\/span><\/span><\/span><\/span><\/span><\/li>\n<li><span class=\"math math-inline\"><span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">[latex]17[\/latex]<\/span><\/span><\/span><\/span><\/span><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p>Therefore, [latex]8+(3\\cdot2^2)-3 = 17[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<p>For some complicated expressions, several passes through the order of operations will be needed. For instance, there may be a radical expression inside parentheses that must be simplified before the parentheses are evaluated. Following the order of operations ensures that anyone simplifying the same mathematical expression will get the same result.<\/p>\n<section class=\"textbox example\">Use the order of operations to evaluate each of the following expressions.<\/p>\n<ol>\n<li>[latex]{\\left(3\\cdot 2\\right)}^{2}-4\\left(6+2\\right)[\/latex]<\/li>\n<li>[latex]\\dfrac{{5}^{2}-4}{7}-\\sqrt{11 - 2}[\/latex]<\/li>\n<li>[latex]6-|5 - 8|+3\\left(4 - 1\\right)[\/latex]<\/li>\n<li>[latex]\\dfrac{14 - 3\\cdot 2}{2\\cdot 5-{3}^{2}}[\/latex]<\/li>\n<li>[latex]7\\left(5\\cdot 3\\right)-2\\left[\\left(6 - 3\\right)-{4}^{2}\\right]+1[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q371324\">Show Solution<\/button><\/p>\n<div id=\"q371324\" class=\"hidden-answer\" style=\"display: none\">\n<p>1.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\left(3\\cdot 2\\right)^{2} & =\\left(6\\right)^{2}-4\\left(8\\right) && \\text{Simplify parentheses} \\\\ & =36-4\\left(8\\right) && \\text{Simplify exponent} \\\\ & =36-32 && \\text{Simplify multiplication} \\\\ & =4 && \\text{Simplify subtraction}\\end{align}[\/latex]<\/p>\n<p>2.<\/p>\n<p class=\"p1\" style=\"text-align: center;\"><span class=\"s1\">[latex]\\begin{align}\\frac{5^{2}-4}{7}-\\sqrt{11-2} & =\\frac{5^{2}-4}{7}-\\sqrt{9} && \\text{Simplify grouping systems (radical)} \\\\ & =\\frac{5^{2}-4}{7}-3 && \\text{Simplify radical} \\\\ & =\\frac{25-4}{7}-3 && \\text{Simplify exponent} \\\\ & =\\frac{21}{7}-3 && \\text{Simplify subtraction in numerator} \\\\ & =3-3 && \\text{Simplify division} \\\\ & =0 && \\text{Simplify subtraction}\\end{align}[\/latex]<\/span><\/p>\n<p>Note that in the first step, the radical is treated as a grouping symbol, like parentheses. Also, in the third step, the fraction bar is considered a grouping symbol so the numerator is considered to be grouped.<\/p>\n<p>3.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}6-|5-8|+3\\left(4-1\\right) & =6-|-3|+3\\left(3\\right) && \\text{Simplify inside grouping system} \\\\ & =6-3+3\\left(3\\right) && \\text{Simplify absolute value} \\\\ & =6-3+9 && \\text{Simplify multiplication} \\\\ & =3+9 && \\text{Simplify subtraction} \\\\ & =12 && \\text{Simplify addition}\\end{align}[\/latex]<\/p>\n<p>4.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\frac{14-3\\cdot2}{2\\cdot5-3^{2}} & =\\frac{14-3\\cdot2}{2\\cdot5-9} && \\text{Simplify exponent} \\\\ & =\\frac{14-6}{10-9} && \\text{Simplify products} \\\\ & =\\frac{8}{1} && \\text{Simplify quotient} \\\\ & =8 && \\text{Simplify quotient}\\end{align}[\/latex]<br \/>\nIn this example, the fraction bar separates the numerator and denominator, which we simplify separately until the last step.<\/p>\n<p>5.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}7\\left(5\\cdot3\\right)-2[\\left(6-3\\right)-4^{2}]+1 & =7\\left(15\\right)-2[\\left(3\\right)-4^{2}]+1 && \\text{Simplify inside parentheses} \\\\ & 7\\left(15\\right)-2\\left(3-16\\right)+1 && \\text{Simplify exponent} \\\\ & =7\\left(15\\right)-2\\left(-13\\right)+1 && \\text{Subtract} \\\\ & =105+26+1 && \\text{Multiply} \\\\ & =132 && \\text{Add}\\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm18712\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=18712&theme=lumen&iframe_resize_id=ohm18712&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm18713\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=18713&theme=lumen&iframe_resize_id=ohm18713&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm18734\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=18734&theme=lumen&iframe_resize_id=ohm18734&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<\/section>\n<div class=\"glossary\"><span class=\"screen-reader-text\" id=\"definition\">definition<\/span><template id=\"term_505_506\"><div class=\"glossary__definition\" role=\"dialog\" data-id=\"term_505_506\"><div tabindex=\"-1\"><p>The exponential notation [latex]a^n[\/latex] means that the number or variable [latex]a[\/latex] is used as a factor [latex]n[\/latex] times.<\/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":6,"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\/505"}],"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":15,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/505\/revisions"}],"predecessor-version":[{"id":4648,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/505\/revisions\/4648"}],"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\/505\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=505"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=505"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=505"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=505"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}