{"id":1125,"date":"2021-10-13T17:47:28","date_gmt":"2021-10-13T17:47:28","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/?post_type=chapter&#038;p=1125"},"modified":"2026-04-01T21:56:53","modified_gmt":"2026-04-01T21:56:53","slug":"1-2-6-order-of-operations","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/chapter\/1-2-6-order-of-operations\/","title":{"raw":"1.2.5: Order of Operations","rendered":"1.2.5: Order of Operations"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h1>Learning Outcomes<\/h1>\r\n<ul>\r\n \t<li>Use the order of operations to simplify arithmetic expressions<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h1>Key words<\/h1>\r\n<ul>\r\n \t<li><strong>Arithmetic operation<\/strong>:\u00a0 [latex]+, -, \\times, \\div, a^{b}, \\sqrt{}[\/latex], etc.<\/li>\r\n \t<li><strong>Arithmetic term<\/strong>: a number, or numbers that are multiplied, divided, or raised to a power<\/li>\r\n \t<li><strong>Arithmetic expression<\/strong>: a term, or terms separated by addition and\/or subtraction<\/li>\r\n \t<li><strong>Order of operations<\/strong>: the order in which arithmetic operations are calculated<\/li>\r\n \t<li><strong>Grouping symbol<\/strong>: symbols used to group terms \u00a0[latex](\\,), [\\, ],\\left\\{\\right\\},\\sqrt{\\, }[\/latex], etc.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Correctly using the order of operations<\/h2>\r\nWhat is [latex]3+5\\times2[\/latex] ? Is it [latex]13[\/latex] or [latex]16[\/latex] ? This may seem like a trick question, but there is actually only one correct answer.\r\n\r\nMany years ago, mathematicians developed a standard <em><b>order of operations<\/b><\/em> that tells us which calculations to make first in an expression with more than one <em><b>operation<\/b><\/em>. In other words, order of operations simply refers to the specific order of steps you should follow when you solve a math expression. Without a standard procedure for making calculations, two people could get two different answers to the same problem, like the one above. So which is it, [latex]13[\/latex] or [latex]16[\/latex] ? By the end of this module you'll know!\r\n\r\nWe need to clarify the order in which operations like [latex]+, -, \\times, \\div, a^{b}, \\sqrt{}[\/latex], etc. will be carried out. Otherwise, expressions may have different meanings, and they may result in different values.\r\n\r\nFor example, consider the expression:\r\n<p style=\"text-align: center;\">[latex]4+3\\cdot 7[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{cccc}\\hfill \\text{Some students say it simplifies to 49.}\\hfill &amp; &amp; &amp; \\hfill \\text{Some students say it simplifies to 25.}\\hfill \\\\ \\begin{array}{ccc}&amp; &amp; \\hfill 4+3\\cdot 7\\hfill \\\\ \\text{Since }4+3\\text{ gives 7.}\\hfill &amp; &amp; \\hfill 7\\cdot 7\\hfill \\\\ \\text{And }7\\cdot 7\\text{ is 49.}\\hfill &amp; &amp; \\hfill 49\\hfill \\end{array}&amp; &amp; &amp; \\begin{array}{ccc}&amp; &amp; \\hfill 4+3\\cdot 7\\hfill \\\\ \\text{Since }3\\cdot 7\\text{ is 21.}\\hfill &amp; &amp; \\hfill 4+21\\hfill \\\\ \\text{And }21+4\\text{ makes 25.}\\hfill &amp; &amp; \\hfill 25\\hfill \\end{array}\\hfill \\end{array}[\/latex]<\/p>\r\nImagine the confusion that could result if every problem had several different correct answers. The same expression should give the same result. So mathematicians established some guidelines called the order of operations, which outlines the order in which parts of an expression must be simplified.\r\n<div class=\"textbox shaded\">\r\n<h3>Order of Operations<\/h3>\r\nWhen simplifying mathematical expressions perform the operations in the following order:\r\n1. Grouping Symbols: parentheses, brackets, braces, etc.\r\n<ul id=\"fs-id1171104029952\">\r\n \t<li>Simplify all expressions inside parentheses or other grouping symbols, working on the innermost parentheses first.<\/li>\r\n<\/ul>\r\n2. Exponents\r\n<ul id=\"fs-id1171104407077\">\r\n \t<li>Simplify all expressions with exponents.<\/li>\r\n<\/ul>\r\n3. Multiplication and Division from left to right\r\n<ul id=\"fs-id1171103140103\">\r\n \t<li>Perform all multiplication and division in order from left to right. These operations have equal priority.<\/li>\r\n<\/ul>\r\n4. Addition and Subtraction\r\n<ul id=\"fs-id1171104002792\">\r\n \t<li>Perform all addition and subtraction in order from left to right. These operations have equal priority.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify the expressions:\r\n<ol>\r\n \t<li>[latex]4+3\\cdot 7[\/latex]<\/li>\r\n \t<li>[latex]\\left(4+3\\right)\\cdot 7[\/latex]<\/li>\r\n<\/ol>\r\nSolution:\r\n<table id=\"eip-id1164750479370\" class=\"unnumbered unstyled\" summary=\"The table shows the expression four plus three times seven. On the next line it states are there any parentheses in the expression? No. The line below that states are there any exponents in the expression? No. The next line states is there any multiplication or division in the expression? Yes. The next line states Multiply first and is followed by the expression of four plus three times seven. The expression is now four plus twenty-one. The last operation is addition. Add four and twenty-one to get twenty-five.\">\r\n<tbody>\r\n<tr>\r\n<th>1.<\/th>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]4+3\\cdot 7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Are there any grouping symbols? No.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Are there any exponents? No.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Is there any multiplication or division? Yes.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply first.<\/td>\r\n<td>[latex]4+\\color{red}{3\\cdot 7}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add.<\/td>\r\n<td>[latex]4+21[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]25[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1164754514704\" class=\"unnumbered unstyled\" summary=\"The image shows the expression four plus three in parentheses times seven. Are there any parentheses? Yes, simplify inside the parentheses by adding four and three to get seven. The expression is now seven times seven. Is there any multiplication or division? Yes, multiply seven by seven to get forty-nine.\">\r\n<tbody>\r\n<tr>\r\n<th>2.<\/th>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex](4+3)\\cdot 7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Are there any parentheses? Yes.<\/td>\r\n<td>[latex]\\color{red}{(4+3)}\\cdot 7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify inside the parentheses.<\/td>\r\n<td>[latex](7)7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Are there any exponents? No.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Is there any multiplication or division? Yes.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]49[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]144748[\/ohm_question]\r\n\r\n[ohm_question]144751[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify:\r\n<ol>\r\n \t<li>[latex]\\text{18}\\div \\text{9}\\cdot \\text{2}[\/latex]<\/li>\r\n \t<li>[latex]\\text{18}\\cdot \\text{9}\\div \\text{2}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"604459\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"604459\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1164754213884\" class=\"unnumbered unstyled\" summary=\"The table shows the expression eighteen divided by nine times two. Are there any parentheses? No. Are there any exponents? No. Is there any multiplication or division? Yes, there is both multiplication and division. Perform multiplication and division from left to right so, divide first. Divide eighteen by nine to get two. The expression is now two times two. The last operation is multiplication. Multiply two by two to get four.\">\r\n<tbody>\r\n<tr>\r\n<th>1.<\/th>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]18\\div 9\\cdot 2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Are there any grouping symbols? No.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Are there any exponents? No.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Is there any multiplication or division? Yes.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply and divide from left to right. Divide.<\/td>\r\n<td>[latex]\\color{red}{2}\\cdot 2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1164752720001\" class=\"unnumbered unstyled\" summary=\"The table shows the expression eighteen times nine divided by two. Are there any parentheses? No. Are there any exponents? No. Is there any multiplication or division? Yes, there is both multiplication and division. Perform multiplication and division from left to right so, multiply first. Multiply eighteen by nine to get one hundred sixty-two. The expression is now one hundred sixty-two divided by two. The last operation is division. divide one hundred sixty-two by two to get eighty-one.\">\r\n<tbody>\r\n<tr>\r\n<th>2.<\/th>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]18\\cdot 9\\div 2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Are there any grouping symbols? No.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Are there any exponents? No.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Is there any multiplication or division? Yes.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply and divide from left to right.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]\\color{red}{162}\\div 2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Divide.<\/td>\r\n<td>[latex]81[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]144756[\/ohm_question]\r\n\r\n<\/div>\r\nAnother way to simplify expressions is to simply each term in an expression to a single number, then add the terms. A\u00a0<em><strong>term<\/strong><\/em> is a number, or numbers that are multiplied, divided, raised to a power, etc., but not added or subtracted.\r\n\r\nFor example, the expression [latex]5^{2}+7\\cdot (-3)+(6\\cdot4)\\div12[\/latex] is an expression with three terms separated by [latex]+[\/latex] signs:\r\n\r\n[latex]5^{2}[\/latex] simplifies to [latex]25[\/latex]\r\n\r\n[latex]7\\cdot (-3)[\/latex] simplifies to [latex]-21[\/latex]\r\n\r\n[latex](6\\cdot4)\\div12[\/latex] simplifies to [latex]24\\div12=2[\/latex].\r\n\r\nThen we add the terms:\u00a0[latex]25+(-21)+2=6[\/latex].\r\n\r\nIf there is a subtraction in the expression, it is best to write the expression as addition of the opposite. That way the subtraction sign gets attached to the term immediately following it. For example. [latex](-4)^{2}-5\\cdot 4=(-4)^{2}+(-5\\cdot 4)=16+(-20)=-4[\/latex].\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]18\\div 6-4\\left(5 - 2\\right)[\/latex].\r\n[reveal-answer q=\"841846\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"841846\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1164754213917\" class=\"unnumbered unstyled\" summary=\"The image shows the expression eighteen divided by six plus four, in parentheses, five minus two. Are there any parentheses? Yes, perform the subtraction inside the parentheses. Five minus two becomes three inside the parentheses. The expression is now eighteen divided by six plus four, in parentheses, three. Are there any exponents? No. Is there any multiplication or division? Yes, there is both multiplication and division. Divide first because multiplication and division are performed left to right. Divide eighteen by six to get three. The expression is now three plus four, in parentheses, three. Now multiply four by the three in parentheses to get twelve. The expression becomes three plus twelve. Add three and twelve to get fifteen.\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 161.421875px;\">Expression<\/th>\r\n<td style=\"width: 286.515625px;\">[latex]18\\div 6\\color{blue}-4(5-2)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 161.421875px;\">1st term<\/td>\r\n<td style=\"width: 286.515625px;\">[latex]\\color{red}{18\\div 6}=3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 161.421875px;\">2nd term<\/td>\r\n<td style=\"width: 286.515625px;\">[latex]-4\\color{blue}{(5-2)}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 161.421875px;\">Parentheses? Yes:<\/td>\r\n<td style=\"width: 286.515625px;\">[latex]\\color{blue}{-4(3)}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 161.421875px;\">Multiply and divide left to right.<\/td>\r\n<td style=\"width: 286.515625px;\">[latex]\\color{blue}-12[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 161.421875px;\">Add terms:<\/td>\r\n<td style=\"width: 286.515625px;\">[latex]\\color{red}{3}+\\color{blue}(-12}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 161.421875px;\">Multiply.<\/td>\r\n<td style=\"width: 286.515625px;\">[latex]-9[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]6\\sqrt{25}+3^{2}\\cdot (-3)-2^{3}\\cdot 4[\/latex]\r\n\r\n[reveal-answer q=\"H76917\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"H76917\"]\r\n\r\nThere are three terms. Use the order of operations on each term.\r\n\r\n[latex]\\;\\;6\\color{red}{\\sqrt{25}}+\\color{blue}{3^{2}}\\cdot (-3)-\\color{green}{2^{3}}\\cdot 4[\/latex]\r\n\r\n[latex]=\\color{red}{6\\cdot 5}+\\color{blue}{9\\cdot (-3)}-\\color{green}{8\\cdot 4}[\/latex]\r\n\r\n[latex]=\\color{red}{30}+\\color{blue}{(-27)}-\\color{green}{32}[\/latex]\r\n\r\n[latex]=-29[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]144758[\/ohm_question]\r\n\r\n<\/div>\r\nIn the video below we show another example of how to use the order of operations to simplify a mathematical expression.\r\n\r\nhttps:\/\/youtu.be\/qFUvF5-w9o0\r\n\r\n&nbsp;\r\n\r\nWhen there are multiple grouping symbols, we simplify the innermost parentheses first and work outward.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\n[latex]\\text{Simplify: }5+{2}^{3}+3\\left[6 - 3\\left(4 - 2\\right)\\right][\/latex].\r\n<h4>Solution<\/h4>\r\nThere are three terms. Use the order of operations on each term.\r\n\r\nColor is used to show exactly what part of the expression is being evaluated at each step.\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}&amp;5+\\color{blue}{{2}^{3}}+3\\left[6 - 3\\color{green}{\\left(4 - 2\\right)}\\right]\\\\&amp;=5+\\color{blue}{8}+3\\left[6 - 3\\color{green}{\\left(2\\right)}\\right]\\\\&amp;=5+8+3\\left[6 - \\color{green}{3\\left(2\\right)}\\right]\\\\&amp;=5+8+3\\left[6 - \\color{green}{6}\\right]\\\\&amp;=5+8+3\\left[6 - \\color{green}{6}\\right]\\\\&amp;=5+8+3\\color{green}{\\left[6 - 6\\right]}\\\\&amp;=5+8+\\color{green}{3\\left[0\\right]}\\\\&amp;=5+8+\\color{green}{0}\\\\&amp;=13\\end{aligned}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]144759[\/ohm_question]\r\n\r\n<\/div>\r\nIn the video below we show another example of how to use the order of operations to simplify an expression that contains exponents and grouping symbols.\r\n\r\nhttps:\/\/youtu.be\/8b-rf2AW3Ac\r\n\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]{2}^{3}+{3}^{4}\\div 3-{5}^{2}[\/latex].\r\n[reveal-answer q=\"199030\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"199030\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1164754324162\" class=\"unnumbered unstyled\" style=\"height: 72px;\" summary=\"The image shows the expression two cubed plus three to the fourth divided by three minus five squared. Are there any parentheses? No. Are there any exponents? Yes, several. Simplify each exponent. Two cubed is eight, three to the fourth is eighty-one, and five squared is twenty-five. The expression becomes eight plus eighty-one divided by three minus twenty-five. Is there any multiplication or division? Yes, just division. Divide eighty-one by three to get twenty-seven. The expression is now eight plus twenty-seven minus five. There is both addition and subtraction left. Perform these from left to right. Eight plus twenty-seven is thirty-five. Now the expression is thirty-five minus twenty five which leaves ten.\">\r\n<tbody>\r\n<tr style=\"height: 12px;\">\r\n<th style=\"width: 173.421875px; height: 12px;\">Expression 3 terms.<\/th>\r\n<th style=\"width: 400.25px; height: 12px;\">[latex]{2}^{3}+{3}^{4}\\div 3-{5}^{2}[\/latex]<\/th>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 173.421875px; height: 12px;\">Simplify exponents in each term.<\/td>\r\n<td style=\"width: 400.25px; height: 12px;\">[latex]\\color{red}{{2}^{3}}+\\color{red}{{3}^{4}}\\div 3-\\color{red}{{5}^{2}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 173.421875px; height: 12px;\">Divide in 2nd term.<\/td>\r\n<td style=\"width: 400.25px; height: 12px;\">[latex]8+\\color{red}{81\\div 3}-25[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 173.421875px; height: 12px;\">Add.<\/td>\r\n<td style=\"width: 400.25px; height: 12px;\">[latex]\\color{red}{8+27}-25[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 173.421875px; height: 12px;\">Subtract.<\/td>\r\n<td style=\"width: 400.25px; height: 12px;\">[latex]\\color{red}{35-25}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 173.421875px; height: 12px;\"><\/td>\r\n<td style=\"width: 400.25px; height: 12px;\">[latex]10[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]144762[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h1>Learning Outcomes<\/h1>\n<ul>\n<li>Use the order of operations to simplify arithmetic expressions<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h1>Key words<\/h1>\n<ul>\n<li><strong>Arithmetic operation<\/strong>:\u00a0 [latex]+, -, \\times, \\div, a^{b}, \\sqrt{}[\/latex], etc.<\/li>\n<li><strong>Arithmetic term<\/strong>: a number, or numbers that are multiplied, divided, or raised to a power<\/li>\n<li><strong>Arithmetic expression<\/strong>: a term, or terms separated by addition and\/or subtraction<\/li>\n<li><strong>Order of operations<\/strong>: the order in which arithmetic operations are calculated<\/li>\n<li><strong>Grouping symbol<\/strong>: symbols used to group terms \u00a0[latex](\\,), [\\, ],\\left\\{\\right\\},\\sqrt{\\, }[\/latex], etc.<\/li>\n<\/ul>\n<\/div>\n<h2>Correctly using the order of operations<\/h2>\n<p>What is [latex]3+5\\times2[\/latex] ? Is it [latex]13[\/latex] or [latex]16[\/latex] ? This may seem like a trick question, but there is actually only one correct answer.<\/p>\n<p>Many years ago, mathematicians developed a standard <em><b>order of operations<\/b><\/em> that tells us which calculations to make first in an expression with more than one <em><b>operation<\/b><\/em>. In other words, order of operations simply refers to the specific order of steps you should follow when you solve a math expression. Without a standard procedure for making calculations, two people could get two different answers to the same problem, like the one above. So which is it, [latex]13[\/latex] or [latex]16[\/latex] ? By the end of this module you&#8217;ll know!<\/p>\n<p>We need to clarify the order in which operations like [latex]+, -, \\times, \\div, a^{b}, \\sqrt{}[\/latex], etc. will be carried out. Otherwise, expressions may have different meanings, and they may result in different values.<\/p>\n<p>For example, consider the expression:<\/p>\n<p style=\"text-align: center;\">[latex]4+3\\cdot 7[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{cccc}\\hfill \\text{Some students say it simplifies to 49.}\\hfill & & & \\hfill \\text{Some students say it simplifies to 25.}\\hfill \\\\ \\begin{array}{ccc}& & \\hfill 4+3\\cdot 7\\hfill \\\\ \\text{Since }4+3\\text{ gives 7.}\\hfill & & \\hfill 7\\cdot 7\\hfill \\\\ \\text{And }7\\cdot 7\\text{ is 49.}\\hfill & & \\hfill 49\\hfill \\end{array}& & & \\begin{array}{ccc}& & \\hfill 4+3\\cdot 7\\hfill \\\\ \\text{Since }3\\cdot 7\\text{ is 21.}\\hfill & & \\hfill 4+21\\hfill \\\\ \\text{And }21+4\\text{ makes 25.}\\hfill & & \\hfill 25\\hfill \\end{array}\\hfill \\end{array}[\/latex]<\/p>\n<p>Imagine the confusion that could result if every problem had several different correct answers. The same expression should give the same result. So mathematicians established some guidelines called the order of operations, which outlines the order in which parts of an expression must be simplified.<\/p>\n<div class=\"textbox shaded\">\n<h3>Order of Operations<\/h3>\n<p>When simplifying mathematical expressions perform the operations in the following order:<br \/>\n1. Grouping Symbols: parentheses, brackets, braces, etc.<\/p>\n<ul id=\"fs-id1171104029952\">\n<li>Simplify all expressions inside parentheses or other grouping symbols, working on the innermost parentheses first.<\/li>\n<\/ul>\n<p>2. Exponents<\/p>\n<ul id=\"fs-id1171104407077\">\n<li>Simplify all expressions with exponents.<\/li>\n<\/ul>\n<p>3. Multiplication and Division from left to right<\/p>\n<ul id=\"fs-id1171103140103\">\n<li>Perform all multiplication and division in order from left to right. These operations have equal priority.<\/li>\n<\/ul>\n<p>4. Addition and Subtraction<\/p>\n<ul id=\"fs-id1171104002792\">\n<li>Perform all addition and subtraction in order from left to right. These operations have equal priority.<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify the expressions:<\/p>\n<ol>\n<li>[latex]4+3\\cdot 7[\/latex]<\/li>\n<li>[latex]\\left(4+3\\right)\\cdot 7[\/latex]<\/li>\n<\/ol>\n<p>Solution:<\/p>\n<table id=\"eip-id1164750479370\" class=\"unnumbered unstyled\" summary=\"The table shows the expression four plus three times seven. On the next line it states are there any parentheses in the expression? No. The line below that states are there any exponents in the expression? No. The next line states is there any multiplication or division in the expression? Yes. The next line states Multiply first and is followed by the expression of four plus three times seven. The expression is now four plus twenty-one. The last operation is addition. Add four and twenty-one to get twenty-five.\">\n<tbody>\n<tr>\n<th>1.<\/th>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]4+3\\cdot 7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Are there any grouping symbols? No.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Are there any exponents? No.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Is there any multiplication or division? Yes.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Multiply first.<\/td>\n<td>[latex]4+\\color{red}{3\\cdot 7}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add.<\/td>\n<td>[latex]4+21[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]25[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1164754514704\" class=\"unnumbered unstyled\" summary=\"The image shows the expression four plus three in parentheses times seven. Are there any parentheses? Yes, simplify inside the parentheses by adding four and three to get seven. The expression is now seven times seven. Is there any multiplication or division? Yes, multiply seven by seven to get forty-nine.\">\n<tbody>\n<tr>\n<th>2.<\/th>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex](4+3)\\cdot 7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Are there any parentheses? Yes.<\/td>\n<td>[latex]\\color{red}{(4+3)}\\cdot 7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify inside the parentheses.<\/td>\n<td>[latex](7)7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Are there any exponents? No.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Is there any multiplication or division? Yes.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]49[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm144748\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144748&theme=oea&iframe_resize_id=ohm144748&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm144751\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144751&theme=oea&iframe_resize_id=ohm144751&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify:<\/p>\n<ol>\n<li>[latex]\\text{18}\\div \\text{9}\\cdot \\text{2}[\/latex]<\/li>\n<li>[latex]\\text{18}\\cdot \\text{9}\\div \\text{2}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q604459\">Show Solution<\/span><\/p>\n<div id=\"q604459\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1164754213884\" class=\"unnumbered unstyled\" summary=\"The table shows the expression eighteen divided by nine times two. Are there any parentheses? No. Are there any exponents? No. Is there any multiplication or division? Yes, there is both multiplication and division. Perform multiplication and division from left to right so, divide first. Divide eighteen by nine to get two. The expression is now two times two. The last operation is multiplication. Multiply two by two to get four.\">\n<tbody>\n<tr>\n<th>1.<\/th>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]18\\div 9\\cdot 2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Are there any grouping symbols? No.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Are there any exponents? No.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Is there any multiplication or division? Yes.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Multiply and divide from left to right. Divide.<\/td>\n<td>[latex]\\color{red}{2}\\cdot 2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]4[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1164752720001\" class=\"unnumbered unstyled\" summary=\"The table shows the expression eighteen times nine divided by two. Are there any parentheses? No. Are there any exponents? No. Is there any multiplication or division? Yes, there is both multiplication and division. Perform multiplication and division from left to right so, multiply first. Multiply eighteen by nine to get one hundred sixty-two. The expression is now one hundred sixty-two divided by two. The last operation is division. divide one hundred sixty-two by two to get eighty-one.\">\n<tbody>\n<tr>\n<th>2.<\/th>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]18\\cdot 9\\div 2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Are there any grouping symbols? No.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Are there any exponents? No.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Is there any multiplication or division? Yes.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Multiply and divide from left to right.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]\\color{red}{162}\\div 2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Divide.<\/td>\n<td>[latex]81[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm144756\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144756&theme=oea&iframe_resize_id=ohm144756&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Another way to simplify expressions is to simply each term in an expression to a single number, then add the terms. A\u00a0<em><strong>term<\/strong><\/em> is a number, or numbers that are multiplied, divided, raised to a power, etc., but not added or subtracted.<\/p>\n<p>For example, the expression [latex]5^{2}+7\\cdot (-3)+(6\\cdot4)\\div12[\/latex] is an expression with three terms separated by [latex]+[\/latex] signs:<\/p>\n<p>[latex]5^{2}[\/latex] simplifies to [latex]25[\/latex]<\/p>\n<p>[latex]7\\cdot (-3)[\/latex] simplifies to [latex]-21[\/latex]<\/p>\n<p>[latex](6\\cdot4)\\div12[\/latex] simplifies to [latex]24\\div12=2[\/latex].<\/p>\n<p>Then we add the terms:\u00a0[latex]25+(-21)+2=6[\/latex].<\/p>\n<p>If there is a subtraction in the expression, it is best to write the expression as addition of the opposite. That way the subtraction sign gets attached to the term immediately following it. For example. [latex](-4)^{2}-5\\cdot 4=(-4)^{2}+(-5\\cdot 4)=16+(-20)=-4[\/latex].<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]18\\div 6-4\\left(5 - 2\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q841846\">Show Solution<\/span><\/p>\n<div id=\"q841846\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1164754213917\" class=\"unnumbered unstyled\" summary=\"The image shows the expression eighteen divided by six plus four, in parentheses, five minus two. Are there any parentheses? Yes, perform the subtraction inside the parentheses. Five minus two becomes three inside the parentheses. The expression is now eighteen divided by six plus four, in parentheses, three. Are there any exponents? No. Is there any multiplication or division? Yes, there is both multiplication and division. Divide first because multiplication and division are performed left to right. Divide eighteen by six to get three. The expression is now three plus four, in parentheses, three. Now multiply four by the three in parentheses to get twelve. The expression becomes three plus twelve. Add three and twelve to get fifteen.\">\n<tbody>\n<tr>\n<th style=\"width: 161.421875px;\">Expression<\/th>\n<td style=\"width: 286.515625px;\">[latex]18\\div 6\\color{blue}-4(5-2)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 161.421875px;\">1st term<\/td>\n<td style=\"width: 286.515625px;\">[latex]\\color{red}{18\\div 6}=3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 161.421875px;\">2nd term<\/td>\n<td style=\"width: 286.515625px;\">[latex]-4\\color{blue}{(5-2)}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 161.421875px;\">Parentheses? Yes:<\/td>\n<td style=\"width: 286.515625px;\">[latex]\\color{blue}{-4(3)}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 161.421875px;\">Multiply and divide left to right.<\/td>\n<td style=\"width: 286.515625px;\">[latex]\\color{blue}-12[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 161.421875px;\">Add terms:<\/td>\n<td style=\"width: 286.515625px;\">[latex]\\color{red}{3}+\\color{blue}(-12}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 161.421875px;\">Multiply.<\/td>\n<td style=\"width: 286.515625px;\">[latex]-9[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]6\\sqrt{25}+3^{2}\\cdot (-3)-2^{3}\\cdot 4[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qH76917\">Show Answer<\/span><\/p>\n<div id=\"qH76917\" class=\"hidden-answer\" style=\"display: none\">\n<p>There are three terms. Use the order of operations on each term.<\/p>\n<p>[latex]\\;\\;6\\color{red}{\\sqrt{25}}+\\color{blue}{3^{2}}\\cdot (-3)-\\color{green}{2^{3}}\\cdot 4[\/latex]<\/p>\n<p>[latex]=\\color{red}{6\\cdot 5}+\\color{blue}{9\\cdot (-3)}-\\color{green}{8\\cdot 4}[\/latex]<\/p>\n<p>[latex]=\\color{red}{30}+\\color{blue}{(-27)}-\\color{green}{32}[\/latex]<\/p>\n<p>[latex]=-29[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm144758\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144758&theme=oea&iframe_resize_id=ohm144758&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the video below we show another example of how to use the order of operations to simplify a mathematical expression.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex: Evaluate an Expression Using the Order of Operations\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/qFUvF5-w9o0?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n<p>When there are multiple grouping symbols, we simplify the innermost parentheses first and work outward.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>[latex]\\text{Simplify: }5+{2}^{3}+3\\left[6 - 3\\left(4 - 2\\right)\\right][\/latex].<\/p>\n<h4>Solution<\/h4>\n<p>There are three terms. Use the order of operations on each term.<\/p>\n<p>Color is used to show exactly what part of the expression is being evaluated at each step.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}&5+\\color{blue}{{2}^{3}}+3\\left[6 - 3\\color{green}{\\left(4 - 2\\right)}\\right]\\\\&=5+\\color{blue}{8}+3\\left[6 - 3\\color{green}{\\left(2\\right)}\\right]\\\\&=5+8+3\\left[6 - \\color{green}{3\\left(2\\right)}\\right]\\\\&=5+8+3\\left[6 - \\color{green}{6}\\right]\\\\&=5+8+3\\left[6 - \\color{green}{6}\\right]\\\\&=5+8+3\\color{green}{\\left[6 - 6\\right]}\\\\&=5+8+\\color{green}{3\\left[0\\right]}\\\\&=5+8+\\color{green}{0}\\\\&=13\\end{aligned}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm144759\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144759&theme=oea&iframe_resize_id=ohm144759&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the video below we show another example of how to use the order of operations to simplify an expression that contains exponents and grouping symbols.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Example 3:  Evaluate An Expression Using The Order of Operation\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/8b-rf2AW3Ac?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]{2}^{3}+{3}^{4}\\div 3-{5}^{2}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q199030\">Show Solution<\/span><\/p>\n<div id=\"q199030\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1164754324162\" class=\"unnumbered unstyled\" style=\"height: 72px;\" summary=\"The image shows the expression two cubed plus three to the fourth divided by three minus five squared. Are there any parentheses? No. Are there any exponents? Yes, several. Simplify each exponent. Two cubed is eight, three to the fourth is eighty-one, and five squared is twenty-five. The expression becomes eight plus eighty-one divided by three minus twenty-five. Is there any multiplication or division? Yes, just division. Divide eighty-one by three to get twenty-seven. The expression is now eight plus twenty-seven minus five. There is both addition and subtraction left. Perform these from left to right. Eight plus twenty-seven is thirty-five. Now the expression is thirty-five minus twenty five which leaves ten.\">\n<tbody>\n<tr style=\"height: 12px;\">\n<th style=\"width: 173.421875px; height: 12px;\">Expression 3 terms.<\/th>\n<th style=\"width: 400.25px; height: 12px;\">[latex]{2}^{3}+{3}^{4}\\div 3-{5}^{2}[\/latex]<\/th>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 173.421875px; height: 12px;\">Simplify exponents in each term.<\/td>\n<td style=\"width: 400.25px; height: 12px;\">[latex]\\color{red}{{2}^{3}}+\\color{red}{{3}^{4}}\\div 3-\\color{red}{{5}^{2}}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 173.421875px; height: 12px;\">Divide in 2nd term.<\/td>\n<td style=\"width: 400.25px; height: 12px;\">[latex]8+\\color{red}{81\\div 3}-25[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 173.421875px; height: 12px;\">Add.<\/td>\n<td style=\"width: 400.25px; height: 12px;\">[latex]\\color{red}{8+27}-25[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 173.421875px; height: 12px;\">Subtract.<\/td>\n<td style=\"width: 400.25px; height: 12px;\">[latex]\\color{red}{35-25}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 173.421875px; height: 12px;\"><\/td>\n<td style=\"width: 400.25px; height: 12px;\">[latex]10[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm144762\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144762&theme=oea&iframe_resize_id=ohm144762&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n","protected":false},"author":422605,"menu_order":7,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1125","chapter","type-chapter","status-publish","hentry"],"part":587,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/1125","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/users\/422605"}],"version-history":[{"count":15,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/1125\/revisions"}],"predecessor-version":[{"id":3224,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/1125\/revisions\/3224"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/parts\/587"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/1125\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/media?parent=1125"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1125"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/contributor?post=1125"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/license?post=1125"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}