{"id":2162,"date":"2021-10-05T15:25:33","date_gmt":"2021-10-05T15:25:33","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/?post_type=chapter&#038;p=2162"},"modified":"2022-03-29T20:37:13","modified_gmt":"2022-03-29T20:37:13","slug":"evaluating-algebraic-expressions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/evaluating-algebraic-expressions\/","title":{"raw":"Evaluating Algebraic Expressions","rendered":"Evaluating Algebraic Expressions"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">Use the order of operations to evaluate algebraic expressions<\/li>\r\n<\/ul>\r\n<\/div>\r\nAn algebraic expression is an expression involving addition, subtraction, multiplication, division, and exponentiation by an exponent that is a rational number. Radicals correspond to fractional exponents, so this definition means we can have integer exponents and radicals.\r\n\r\nWhen you apply the order of operations to an expression involving a fraction, remember there are implied grouping symbols around the terms in the numerator and denominator.\r\n\r\nFor example, [latex]\\frac{1+2}{3+4}[\/latex] really means [latex]\\frac{(1+2)}{(3+4)}[\/latex]. Since we perform operations within grouping symbols first, [latex]\\frac{1+2}{3+4} = \\frac{3}{7}[\/latex].\u00a0To find the decimal representation of this number using a calculator, you can simplify the numerator and denominator and then divide,\r\n<p style=\"text-align: center;\">[latex]\\frac{1+2}{3+4} = \\frac{3}{7} \\approx 0.4286[\/latex].<\/p>\r\nOr, you can enter the original expression, inserting parentheses around the numerator and the denominator,\r\n<p style=\"text-align: center;\">[latex](1+2) \\div (3+4) \\approx 0.4286[\/latex].<\/p>\r\nIn Module 10 you will encounter some complicated expressions involving fractions and radicals. In this section, we\u2019ll review how to evaluate expressions like these.\r\n<h2>Complex Fractions<\/h2>\r\nA <strong>complex fraction<\/strong> is a fraction which contains fractions in the numerator and\/or the denominator. For example, [latex]\\frac{1+ \\frac{1}{2}}{1- \\frac{1}{4}}[\/latex], is a complex fraction.\u00a0The fractions in the numerator and denominator are called <strong>secondary fractions<\/strong>. To evaluate a complex fraction, you can simplify the numerator and denominator separately and then divide.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nEvaluate: [latex]\\frac{1+ \\frac{1}{2}}{1- \\frac{1}{4}}[\/latex].\r\n\r\n[reveal-answer q=\"588177\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"588177\"]\r\n\r\nSimplify the numerator: [latex]1+\\frac{1}{2} = \\frac{2}{2} + \\frac{1}{2} = \\frac{2+1}{2} = \\frac{3}{2}[\/latex].\r\n\r\nSimplify the denominator: [latex]1- \\frac{1}{4} = \\frac{4}{4} - \\frac{1}{4} = \\frac{4-1}{4} = \\frac{3}{4}[\/latex].\r\n\r\nDivide: [latex]\\frac{1+ \\frac{1}{2}}{1- \\frac{1}{4}} = \\frac{\\frac{3}{2}}{\\frac{3}{4}} = \\frac{3}{2} \\cdot \\frac{4}{3} = \\frac{3 \\cdot 2 \\cdot 2}{2 \\cdot 3} = 2[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nTo evaluate [latex]\\frac{1+ \\frac{1}{2}}{1- \\frac{1}{4}}[\/latex]\u00a0on a calculator, as before, we insert parentheses around the numerator and denominator and divide:\r\n<p style=\"text-align: center;\">[latex](1+2\u00f72)\u00f7(1-1\u00f74)=2[\/latex].<\/p>\r\n\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nEvaluate: [latex]\\frac{\\frac{2}{3} + \\frac{1}{6}}{\\frac{1}{2} + \\frac{1}{4}}[\/latex].\r\n\r\n[reveal-answer q=\"943246\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"943246\"]\r\n\r\nSimplify the numerator: [latex]\\frac{2}{3} + \\frac{1}{6} = \\frac{4}{6} + \\frac{1}{6} = \\frac{4+1}{6} = \\frac{5}{6}[\/latex].\r\n\r\nSimplify the denominator: [latex]\\frac{1}{2} + \\frac{1}{4} = \\frac{2}{4} + \\frac{1}{4} = \\frac{2+1}{4} = \\frac{3}{4}[\/latex].\r\n\r\nDivide: [latex]\\frac{\\frac{2}{3} + \\frac{1}{6}}{\\frac{1}{2} + \\frac{1}{4}} = \\frac{\\frac{5}{6}}{\\frac{3}{4}} = \\frac{5}{6} \\cdot \\frac{4}{3} = \\frac{5 \\cdot 2 \\cdot 2}{2 \\cdot 3 \\cdot 3} = \\frac{5 \\cdot 2}{3 \\cdot 3} = \\frac{10}{9} \\approx 1.1111[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nTo evaluate the expression directly using a calculator,\r\n<p style=\"text-align: center;\">[latex]\\frac{\\frac{2}{3} + \\frac{1}{6}}{\\frac{1}{2} + \\frac{1}{4}} = (2 \\div 3+1 \\div 4) \\div (1 \\div 2+1 \\div 4) \\approx 1.1111[\/latex].<\/p>\r\n\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nEvaluate: [latex]\\frac{\\frac{3^2}{5}}{\\frac{2^2}{3}}[\/latex].\r\n\r\n[reveal-answer q=\"266792\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"266792\"]\r\n\r\nSimplify the numerator: [latex]\\frac{3^2}{5} = \\frac{9}{5}[\/latex].\r\n\r\nSimplify the denominator: [latex]\\frac{2^2}{3} = \\frac{3}{4}[\/latex].\r\n\r\nDivide: [latex]\\frac{\\frac{9}{5}}{\\frac{4}{3}} = \\frac{9}{5} \\cdot \\frac{3}{4} = \\frac{27}{20} = 1.35[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nTo evaluate the expression directly using a calculator, enter\r\n<p style=\"text-align: center;\">[latex](3^2 \\div 5) \\div (2^2 \\div 3) = 1.35[\/latex]<\/p>\r\n\r\n<h2>Expressions Involving Fractions and Radicals<\/h2>\r\nWhen you found the test statistic for a hypothesis test for means,\r\n<p style=\"text-align: center;\">[latex]\\large \\frac{\\overline{x} - \\mu _\\overline{x}}{\\sigma _ \\overline{x}}[\/latex],<\/p>\r\nthe formula for the standard error was [latex]\\sigma _\\overline{x} = \\frac{\\sigma _x}{\\sqrt{n}}[\/latex]. So we could also have written the test statistics as\r\n<p style=\"text-align: center;\">[latex]\\large \\frac{\\overline{x} - \\mu _\\overline{x}}{\\frac{\\sigma _x}{\\sqrt{n}}}[\/latex].<\/p>\r\nThis can be evaluated directly on a calculator as long as parentheses are inserted properly.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSuppose that the weights of the contents of cereal boxes are normally distributed with a mean of 16 ounces and a standard deviation of 0.25 ounces. In a random sample of 36 boxes of cereal, the mean weight is observed to be 15.8 ounces. Find the value of the test statistic.\r\n\r\n[reveal-answer q=\"256405\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"256405\"]\r\n<p style=\"text-align: center;\">[latex]\\large \\frac{\\overline{x} - \\mu _\\overline{x}}{\\frac{\\sigma _x}{\\sqrt{n}}} = \\frac{15.8-16}{\\frac{0.25}{\\sqrt{36}}} = \\frac{-0.2}{\\frac{0.5}{6}} = \\frac{-0.2}{1} \\cdot \\frac{6}{0.5} = \\frac{-1.2}{0.5} = -2.4[\/latex]<\/p>\r\nIt is much faster to evaluate this expression using a calculator with one line entry.\r\n<p style=\"text-align: center;\">[latex]\\large \\frac{\\overline{x} - \\mu _\\overline{x}}{\\frac{\\sigma _x}{\\sqrt{n}}} =\\frac{15.8-16}{\\frac{0.25}{\\sqrt{36}}} = (15.8-16) \\div (0.25 \\div \\sqrt{36} ) = -2.4[\/latex]<\/p>\r\nNote that many calculators insert a left parenthesis when you use the square root function. Therefore, your entry might look like this:\r\n<p style=\"text-align: center;\">[latex]\\large \\frac{15.8-16}{\\frac{0.25}{\\sqrt{36}}} =(15.8-16) \\div (0.25 \\div \\sqrt{} (36) ) = -2.4[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nEvaluate [latex]\\large \\frac{4.5-5}{\\sqrt{\\frac{2^2}{25} + \\frac{3^2}{16}}}[\/latex] using a calculator.\u00a0Round your answer to four decimal places if necessary.\r\n\r\n[reveal-answer q=\"74805\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"74805\"]\r\n\r\nMake sure you enter parentheses around the two terms in the numerator. The radical is an implied grouping.\r\n<p style=\"text-align: center;\">[latex](4.5-5) \\div \\sqrt{(2^2 \\div 25 +3^2 \\div 16)} = -0.5882[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nEvaluate [latex]\\large \\frac{0.5-0.4}{\\sqrt{0.45(1-0.45)(\\frac{1}{100} + \\frac{1}{100})}}[\/latex] using a calculator.\r\n\r\n[reveal-answer q=\"970849\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"970849\"]\r\n\r\nMake sure you enter parentheses around the two terms in the numerator. The radical is an implied grouping.\r\n<p style=\"text-align: center;\">[latex](0.5-0.4) \\div \\sqrt{(0.45 \\cdot (1-0.45) \\cdot (1 \\div 100 + 1 \\div 100))} = 1.4213[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\">Use the order of operations to evaluate algebraic expressions<\/li>\n<\/ul>\n<\/div>\n<p>An algebraic expression is an expression involving addition, subtraction, multiplication, division, and exponentiation by an exponent that is a rational number. Radicals correspond to fractional exponents, so this definition means we can have integer exponents and radicals.<\/p>\n<p>When you apply the order of operations to an expression involving a fraction, remember there are implied grouping symbols around the terms in the numerator and denominator.<\/p>\n<p>For example, [latex]\\frac{1+2}{3+4}[\/latex] really means [latex]\\frac{(1+2)}{(3+4)}[\/latex]. Since we perform operations within grouping symbols first, [latex]\\frac{1+2}{3+4} = \\frac{3}{7}[\/latex].\u00a0To find the decimal representation of this number using a calculator, you can simplify the numerator and denominator and then divide,<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{1+2}{3+4} = \\frac{3}{7} \\approx 0.4286[\/latex].<\/p>\n<p>Or, you can enter the original expression, inserting parentheses around the numerator and the denominator,<\/p>\n<p style=\"text-align: center;\">[latex](1+2) \\div (3+4) \\approx 0.4286[\/latex].<\/p>\n<p>In Module 10 you will encounter some complicated expressions involving fractions and radicals. In this section, we\u2019ll review how to evaluate expressions like these.<\/p>\n<h2>Complex Fractions<\/h2>\n<p>A <strong>complex fraction<\/strong> is a fraction which contains fractions in the numerator and\/or the denominator. For example, [latex]\\frac{1+ \\frac{1}{2}}{1- \\frac{1}{4}}[\/latex], is a complex fraction.\u00a0The fractions in the numerator and denominator are called <strong>secondary fractions<\/strong>. To evaluate a complex fraction, you can simplify the numerator and denominator separately and then divide.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Evaluate: [latex]\\frac{1+ \\frac{1}{2}}{1- \\frac{1}{4}}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q588177\">Show Answer<\/span><\/p>\n<div id=\"q588177\" class=\"hidden-answer\" style=\"display: none\">\n<p>Simplify the numerator: [latex]1+\\frac{1}{2} = \\frac{2}{2} + \\frac{1}{2} = \\frac{2+1}{2} = \\frac{3}{2}[\/latex].<\/p>\n<p>Simplify the denominator: [latex]1- \\frac{1}{4} = \\frac{4}{4} - \\frac{1}{4} = \\frac{4-1}{4} = \\frac{3}{4}[\/latex].<\/p>\n<p>Divide: [latex]\\frac{1+ \\frac{1}{2}}{1- \\frac{1}{4}} = \\frac{\\frac{3}{2}}{\\frac{3}{4}} = \\frac{3}{2} \\cdot \\frac{4}{3} = \\frac{3 \\cdot 2 \\cdot 2}{2 \\cdot 3} = 2[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>To evaluate [latex]\\frac{1+ \\frac{1}{2}}{1- \\frac{1}{4}}[\/latex]\u00a0on a calculator, as before, we insert parentheses around the numerator and denominator and divide:<\/p>\n<p style=\"text-align: center;\">[latex](1+2\u00f72)\u00f7(1-1\u00f74)=2[\/latex].<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Evaluate: [latex]\\frac{\\frac{2}{3} + \\frac{1}{6}}{\\frac{1}{2} + \\frac{1}{4}}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q943246\">Show Answer<\/span><\/p>\n<div id=\"q943246\" class=\"hidden-answer\" style=\"display: none\">\n<p>Simplify the numerator: [latex]\\frac{2}{3} + \\frac{1}{6} = \\frac{4}{6} + \\frac{1}{6} = \\frac{4+1}{6} = \\frac{5}{6}[\/latex].<\/p>\n<p>Simplify the denominator: [latex]\\frac{1}{2} + \\frac{1}{4} = \\frac{2}{4} + \\frac{1}{4} = \\frac{2+1}{4} = \\frac{3}{4}[\/latex].<\/p>\n<p>Divide: [latex]\\frac{\\frac{2}{3} + \\frac{1}{6}}{\\frac{1}{2} + \\frac{1}{4}} = \\frac{\\frac{5}{6}}{\\frac{3}{4}} = \\frac{5}{6} \\cdot \\frac{4}{3} = \\frac{5 \\cdot 2 \\cdot 2}{2 \\cdot 3 \\cdot 3} = \\frac{5 \\cdot 2}{3 \\cdot 3} = \\frac{10}{9} \\approx 1.1111[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>To evaluate the expression directly using a calculator,<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{\\frac{2}{3} + \\frac{1}{6}}{\\frac{1}{2} + \\frac{1}{4}} = (2 \\div 3+1 \\div 4) \\div (1 \\div 2+1 \\div 4) \\approx 1.1111[\/latex].<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Evaluate: [latex]\\frac{\\frac{3^2}{5}}{\\frac{2^2}{3}}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q266792\">Show Answer<\/span><\/p>\n<div id=\"q266792\" class=\"hidden-answer\" style=\"display: none\">\n<p>Simplify the numerator: [latex]\\frac{3^2}{5} = \\frac{9}{5}[\/latex].<\/p>\n<p>Simplify the denominator: [latex]\\frac{2^2}{3} = \\frac{3}{4}[\/latex].<\/p>\n<p>Divide: [latex]\\frac{\\frac{9}{5}}{\\frac{4}{3}} = \\frac{9}{5} \\cdot \\frac{3}{4} = \\frac{27}{20} = 1.35[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>To evaluate the expression directly using a calculator, enter<\/p>\n<p style=\"text-align: center;\">[latex](3^2 \\div 5) \\div (2^2 \\div 3) = 1.35[\/latex]<\/p>\n<h2>Expressions Involving Fractions and Radicals<\/h2>\n<p>When you found the test statistic for a hypothesis test for means,<\/p>\n<p style=\"text-align: center;\">[latex]\\large \\frac{\\overline{x} - \\mu _\\overline{x}}{\\sigma _ \\overline{x}}[\/latex],<\/p>\n<p>the formula for the standard error was [latex]\\sigma _\\overline{x} = \\frac{\\sigma _x}{\\sqrt{n}}[\/latex]. So we could also have written the test statistics as<\/p>\n<p style=\"text-align: center;\">[latex]\\large \\frac{\\overline{x} - \\mu _\\overline{x}}{\\frac{\\sigma _x}{\\sqrt{n}}}[\/latex].<\/p>\n<p>This can be evaluated directly on a calculator as long as parentheses are inserted properly.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Suppose that the weights of the contents of cereal boxes are normally distributed with a mean of 16 ounces and a standard deviation of 0.25 ounces. In a random sample of 36 boxes of cereal, the mean weight is observed to be 15.8 ounces. Find the value of the test statistic.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q256405\">Show Answer<\/span><\/p>\n<div id=\"q256405\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\">[latex]\\large \\frac{\\overline{x} - \\mu _\\overline{x}}{\\frac{\\sigma _x}{\\sqrt{n}}} = \\frac{15.8-16}{\\frac{0.25}{\\sqrt{36}}} = \\frac{-0.2}{\\frac{0.5}{6}} = \\frac{-0.2}{1} \\cdot \\frac{6}{0.5} = \\frac{-1.2}{0.5} = -2.4[\/latex]<\/p>\n<p>It is much faster to evaluate this expression using a calculator with one line entry.<\/p>\n<p style=\"text-align: center;\">[latex]\\large \\frac{\\overline{x} - \\mu _\\overline{x}}{\\frac{\\sigma _x}{\\sqrt{n}}} =\\frac{15.8-16}{\\frac{0.25}{\\sqrt{36}}} = (15.8-16) \\div (0.25 \\div \\sqrt{36} ) = -2.4[\/latex]<\/p>\n<p>Note that many calculators insert a left parenthesis when you use the square root function. Therefore, your entry might look like this:<\/p>\n<p style=\"text-align: center;\">[latex]\\large \\frac{15.8-16}{\\frac{0.25}{\\sqrt{36}}} =(15.8-16) \\div (0.25 \\div \\sqrt{} (36) ) = -2.4[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Evaluate [latex]\\large \\frac{4.5-5}{\\sqrt{\\frac{2^2}{25} + \\frac{3^2}{16}}}[\/latex] using a calculator.\u00a0Round your answer to four decimal places if necessary.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q74805\">Show Answer<\/span><\/p>\n<div id=\"q74805\" class=\"hidden-answer\" style=\"display: none\">\n<p>Make sure you enter parentheses around the two terms in the numerator. The radical is an implied grouping.<\/p>\n<p style=\"text-align: center;\">[latex](4.5-5) \\div \\sqrt{(2^2 \\div 25 +3^2 \\div 16)} = -0.5882[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Evaluate [latex]\\large \\frac{0.5-0.4}{\\sqrt{0.45(1-0.45)(\\frac{1}{100} + \\frac{1}{100})}}[\/latex] using a calculator.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q970849\">Show Answer<\/span><\/p>\n<div id=\"q970849\" class=\"hidden-answer\" style=\"display: none\">\n<p>Make sure you enter parentheses around the two terms in the numerator. The radical is an implied grouping.<\/p>\n<p style=\"text-align: center;\">[latex](0.5-0.4) \\div \\sqrt{(0.45 \\cdot (1-0.45) \\cdot (1 \\div 100 + 1 \\div 100))} = 1.4213[\/latex]<\/p>\n<\/div>\n<\/div>\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-2162\">\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><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><\/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":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2162","chapter","type-chapter","status-publish","hentry"],"part":285,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/2162","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/users\/169134"}],"version-history":[{"count":28,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/2162\/revisions"}],"predecessor-version":[{"id":3917,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/2162\/revisions\/3917"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/parts\/285"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/2162\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/media?parent=2162"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=2162"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/contributor?post=2162"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/license?post=2162"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}