{"id":522,"date":"2021-08-02T19:55:33","date_gmt":"2021-08-02T19:55:33","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/?post_type=chapter&#038;p=522"},"modified":"2022-01-18T19:47:17","modified_gmt":"2022-01-18T19:47:17","slug":"evaluating-expressions-with-real-numbers","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/evaluating-expressions-with-real-numbers\/","title":{"raw":"Evaluating Expressions With Real Numbers","rendered":"Evaluating Expressions With Real Numbers"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Use the order of operations to evaluate expressions with real numbers<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Introduction<\/h2>\r\nLet's review some important terminology before we begin:\r\n\r\nA mathematical <strong>expression<\/strong> combines numbers and variables with mathematical operations such as addition, subtraction, multiplication, and addition. For example, [latex]2+8 \\cdot 5[\/latex] is an expression.\r\n\r\nWe obtain different results depending on the order in which we perform the operations in an expression. We need a set of conventions so that everyone arrives at the same value when evaluating an expression.\r\n<h2>Order of Operations<\/h2>\r\n<ul>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">Perform all operations within grouping symbols first. Grouping symbols include ( ), [ ],\u00a0 and { }.\u00a0 If there are nested groupings, evaluate within the innermost grouping symbols first.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">Evaluate exponents and radicals (such as square roots)<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">Multiply and divide, left to right<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">Add and subtract, left to right<\/li>\r\n<\/ul>\r\nA helpful acronym for remembering the order of operations is <strong>PEMDAS<\/strong> (parenthesis, exponents, multiplications and divisions, additions, and subtractions).\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify [latex]2 + 8 \\cdot 5[\/latex].\r\n\r\n[reveal-answer q=\"689364\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"689364\"]\r\n\r\nSince multiplications are performed before addition, we have\r\n\r\n[latex]2+8 \\cdot 5 = 2+40 = 42[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify [latex]10-6 \\div 3 \\cdot 2 +1[\/latex].\r\n\r\n[reveal-answer q=\"944910\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"944910\"]\r\n\r\nFirst, we perform multiplications and divisions left to right.\r\n\r\n[latex]10-6 \\div 3 \\cdot 2 + 1 = 10 - 2 \\cdot 2 + 1 = 10 - 4 + 1[\/latex]\r\n\r\nNext, we perform addition and subtractions from left to right.\r\n\r\n[latex] 10 - 4 + 1 = 6 + 1 = 7[\/latex]\r\n\r\nTherefore,\u00a0 [latex]10 - 6 \\div 3 \\cdot 2 + 1 = 7[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\nIn the following example, you will be shown how to simplify an expression that contains both multiplication and subtraction using the order of operations.\r\n\r\nhttps:\/\/youtu.be\/yqp06obmcVc\r\n<h2>Exponents<\/h2>\r\nWhen you are evaluating expressions, you will sometimes see exponents used to represent repeated multiplication. Recall that an expression such as [latex]2^{3}[\/latex] is <strong>exponential notation<\/strong> for [latex]2 \\cdot 2 \\cdot 2[\/latex]. Exponential notation has two parts: the <strong>base<\/strong> and the <strong>exponent<\/strong> or the <strong>power<\/strong>. In [latex]2^{3}, 2[\/latex] is the base, and [latex]3[\/latex] is the exponent. The exponent determines how many times the base is multiplied by itself.\r\n\r\nExponents are a way to represent repeated multiplication; the order of operations places it <em>before<\/em> any other multiplication, division, subtraction, and addition is performed.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify [latex]5 \\cdot 2^{3}[\/latex]\r\n\r\n[reveal-answer q=\"699864\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"699864\"]\r\n\r\nWorking according to the order of operations, we evaluate the exponential expression first and then multiply.\r\n\r\n[latex]5 \\cdot 2^{3} = 5 \\cdot 8 = 40[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the video that follows, an expression with exponents on its terms is simplified using the order of operations.\r\n\r\nhttps:\/\/youtu.be\/JjBBgV7G_Qw\r\n<h2>Grouping Symbols<\/h2>\r\nGrouping symbols such as parentheses ( ), brackets [ ], braces{ }, and fraction bars can be used to further control the order of the four arithmetic operations. The rules of the order of operations require computations within grouping symbols to be completed first. As you evaluate within a grouping symbol, follow the order of operations. After computing within the grouping symbols, divide or multiply from left to right and then subtract or add from left to right. When there are grouping symbols within grouping symbols, calculate from the inside to the outside. That is, begin simplifying within the innermost grouping symbols first.\r\n\r\nGrouping symbols can also be used to show multiplication. In the example that follows, both uses of parentheses\u2014as a way to represent a group, as well as a way to express multiplication\u2014are shown.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify [latex][12-(1+3^{2})](5)[\/latex]\r\n\r\n[reveal-answer q=\"394068\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"394068\"]\r\n\r\nWe begin by evaluating within the innermost set of grouping symbols, working according to the order of operations.\r\n\r\n[latex][12-(1+3^{2})](5)=[12-(1+9)](5)=[12-(10)](5)=(2)(5)=10[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSimplify [latex] \\frac{2+7}{4-1}[\/latex]\r\n\r\n[reveal-answer q=\"575194\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"575194\"]\r\n\r\nFirst we perform the operations in the numerator and the operations in the denominator separately.\r\n\r\n[latex]\\frac{2+7}{4-1} = \\frac{9}{3}[\/latex]\r\n\r\nThe fraction bar represents division, so our final result is\r\n\r\n[latex]\\frac{9}{3}=3[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video, you are shown how to use the order of operations to simplify an expression with grouping symbols, exponents, multiplication, and addition.\r\n\r\nhttps:\/\/youtu.be\/EMch2MKCVdA\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]389-390-453[\/ohm_question]\r\n\r\n<\/div>\r\nOften we will need to evaluate a formula for given values of the variables. When we do so, we replace each variable with its given value and evaluate according to the order of operations. For example, the volume of a square pyramid can be found by evaluating [latex]V=\\frac{1}{3}s^{2}h[\/latex], where [latex]V[\/latex] represents volume, [latex]s[\/latex] is the length of the sides of the square base, and [latex]h[\/latex] is the height. If [latex]s=12[\/latex] centimeters and [latex]h=10[\/latex] centimeters, the volume of the square pyramid is\r\n\r\n[latex]V=\\frac{1}{3} \\cdot 12^{2} \\cdot 10 = \\frac{1}{3} \\cdot 144 \\cdot 10 = \\frac{144}{3} \\cdot 10 = 48 \\cdot 10 = 480[\/latex] cubic centimeters.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]2376[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Use the order of operations to evaluate expressions with real numbers<\/li>\n<\/ul>\n<\/div>\n<h2>Introduction<\/h2>\n<p>Let&#8217;s review some important terminology before we begin:<\/p>\n<p>A mathematical <strong>expression<\/strong> combines numbers and variables with mathematical operations such as addition, subtraction, multiplication, and addition. For example, [latex]2+8 \\cdot 5[\/latex] is an expression.<\/p>\n<p>We obtain different results depending on the order in which we perform the operations in an expression. We need a set of conventions so that everyone arrives at the same value when evaluating an expression.<\/p>\n<h2>Order of Operations<\/h2>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\">Perform all operations within grouping symbols first. Grouping symbols include ( ), [ ],\u00a0 and { }.\u00a0 If there are nested groupings, evaluate within the innermost grouping symbols first.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">Evaluate exponents and radicals (such as square roots)<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">Multiply and divide, left to right<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">Add and subtract, left to right<\/li>\n<\/ul>\n<p>A helpful acronym for remembering the order of operations is <strong>PEMDAS<\/strong> (parenthesis, exponents, multiplications and divisions, additions, and subtractions).<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify [latex]2 + 8 \\cdot 5[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q689364\">Show Answer<\/span><\/p>\n<div id=\"q689364\" class=\"hidden-answer\" style=\"display: none\">\n<p>Since multiplications are performed before addition, we have<\/p>\n<p>[latex]2+8 \\cdot 5 = 2+40 = 42[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify [latex]10-6 \\div 3 \\cdot 2 +1[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q944910\">Show Answer<\/span><\/p>\n<div id=\"q944910\" class=\"hidden-answer\" style=\"display: none\">\n<p>First, we perform multiplications and divisions left to right.<\/p>\n<p>[latex]10-6 \\div 3 \\cdot 2 + 1 = 10 - 2 \\cdot 2 + 1 = 10 - 4 + 1[\/latex]<\/p>\n<p>Next, we perform addition and subtractions from left to right.<\/p>\n<p>[latex]10 - 4 + 1 = 6 + 1 = 7[\/latex]<\/p>\n<p>Therefore,\u00a0 [latex]10 - 6 \\div 3 \\cdot 2 + 1 = 7[\/latex]<\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<p>In the following example, you will be shown how to simplify an expression that contains both multiplication and subtraction using the order of operations.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Simplify an Expression in the Form:  a*1\/b-c\/(1\/d)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/yqp06obmcVc?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Exponents<\/h2>\n<p>When you are evaluating expressions, you will sometimes see exponents used to represent repeated multiplication. Recall that an expression such as [latex]2^{3}[\/latex] is <strong>exponential notation<\/strong> for [latex]2 \\cdot 2 \\cdot 2[\/latex]. Exponential notation has two parts: the <strong>base<\/strong> and the <strong>exponent<\/strong> or the <strong>power<\/strong>. In [latex]2^{3}, 2[\/latex] is the base, and [latex]3[\/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 <em>before<\/em> any other multiplication, division, subtraction, and addition is performed.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify [latex]5 \\cdot 2^{3}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q699864\">Show Answer<\/span><\/p>\n<div id=\"q699864\" class=\"hidden-answer\" style=\"display: none\">\n<p>Working according to the order of operations, we evaluate the exponential expression first and then multiply.<\/p>\n<p>[latex]5 \\cdot 2^{3} = 5 \\cdot 8 = 40[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the video that follows, an expression with exponents on its terms is simplified using the order of operations.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Simplify an Expression in the Form:  a^n*b^m\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/JjBBgV7G_Qw?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Grouping Symbols<\/h2>\n<p>Grouping symbols such as parentheses ( ), brackets [ ], braces{ }, and fraction bars can be used to further control the order of the four arithmetic operations. The rules of the order of operations require computations within grouping symbols to be completed first. As you evaluate within a grouping symbol, follow the order of operations. After computing within the grouping symbols, divide or multiply from left to right and then subtract or add from left to right. When there are grouping symbols within grouping symbols, calculate from the inside to the outside. That is, begin simplifying within the innermost grouping symbols first.<\/p>\n<p>Grouping symbols can also be used to show multiplication. In the example that follows, both uses of parentheses\u2014as a way to represent a group, as well as a way to express multiplication\u2014are shown.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify [latex][12-(1+3^{2})](5)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q394068\">Show Answer<\/span><\/p>\n<div id=\"q394068\" class=\"hidden-answer\" style=\"display: none\">\n<p>We begin by evaluating within the innermost set of grouping symbols, working according to the order of operations.<\/p>\n<p>[latex][12-(1+3^{2})](5)=[12-(1+9)](5)=[12-(10)](5)=(2)(5)=10[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Simplify [latex]\\frac{2+7}{4-1}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q575194\">Show Answer<\/span><\/p>\n<div id=\"q575194\" class=\"hidden-answer\" style=\"display: none\">\n<p>First we perform the operations in the numerator and the operations in the denominator separately.<\/p>\n<p>[latex]\\frac{2+7}{4-1} = \\frac{9}{3}[\/latex]<\/p>\n<p>The fraction bar represents division, so our final result is<\/p>\n<p>[latex]\\frac{9}{3}=3[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video, you are shown how to use the order of operations to simplify an expression with grouping symbols, exponents, multiplication, and addition.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Simplify an Expression in the Form:  (a+b)^2+c*d\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/EMch2MKCVdA?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm389\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=389-390-453&theme=oea&iframe_resize_id=ohm389&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Often we will need to evaluate a formula for given values of the variables. When we do so, we replace each variable with its given value and evaluate according to the order of operations. For example, the volume of a square pyramid can be found by evaluating [latex]V=\\frac{1}{3}s^{2}h[\/latex], where [latex]V[\/latex] represents volume, [latex]s[\/latex] is the length of the sides of the square base, and [latex]h[\/latex] is the height. If [latex]s=12[\/latex] centimeters and [latex]h=10[\/latex] centimeters, the volume of the square pyramid is<\/p>\n<p>[latex]V=\\frac{1}{3} \\cdot 12^{2} \\cdot 10 = \\frac{1}{3} \\cdot 144 \\cdot 10 = \\frac{144}{3} \\cdot 10 = 48 \\cdot 10 = 480[\/latex] cubic centimeters.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm2376\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2376&theme=oea&iframe_resize_id=ohm2376&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-522\">\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>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>QID 389, 390, 453, 2376. <strong>Authored by<\/strong>: Lippman, D. <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>. <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>Simplify an Expression in the Form: a*1\/b-c\/(1\/d). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/yqp06obmcVc\">https:\/\/youtu.be\/yqp06obmcVc<\/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>Simplify an Expression in the Form: a^n*b^m. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/JjBBgV7G_Qw\">https:\/\/youtu.be\/JjBBgV7G_Qw<\/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>Simplify an Expression in the Form: (a+b)^2+c*d. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/EMch2MKCVdA\">https:\/\/youtu.be\/EMch2MKCVdA<\/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>Unit 9: Real Numbers, 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\/dm-http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/\">http:\/\/nrocnetwork.org\/dm-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":169134,"menu_order":5,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Simplify an Expression in the Form: a*1\/b-c\/(1\/d)\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen 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