{"id":9207,"date":"2017-05-02T16:40:44","date_gmt":"2017-05-02T16:40:44","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/prealgebra\/?post_type=chapter&#038;p=9207"},"modified":"2019-01-28T16:18:39","modified_gmt":"2019-01-28T16:18:39","slug":"simplifying-expressions-using-the-order-of-operations","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/csn-prealgebra\/chapter\/simplifying-expressions-using-the-order-of-operations\/","title":{"raw":"Simplifying Expressions Using the Order of Operations","rendered":"Simplifying Expressions Using the Order of Operations"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Use the order of operations to simplify mathematical expressions<\/li>\r\n \t<li>Simplify mathematical expressions involving addition, subtraction, multiplication, division, and exponents<\/li>\r\n<\/ul>\r\n<\/div>\r\n&nbsp;\r\n<h2>Simplify Expressions Using the Order of Operations<\/h2>\r\nWe\u2019ve introduced most of the symbols and notation used in algebra, but now we need to clarify the order of operations. 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. <strong>P<\/strong>arentheses and other Grouping Symbols\r\n<ul id=\"fs-id1171104029952\">\r\n \t<li>Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first.<\/li>\r\n<\/ul>\r\n2. <strong>E<\/strong>xponents\r\n<ul id=\"fs-id1171104407077\">\r\n \t<li>Simplify all expressions with exponents.<\/li>\r\n<\/ul>\r\n3. <strong>M<\/strong>ultiplication and <strong>D<\/strong>ivision\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. <strong>A<\/strong>ddition and <strong>S<\/strong>ubtraction\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&nbsp;\r\n\r\nStudents often ask, \"How will I remember the order?\" Here is a way to help you remember: Take the first letter of each key word and substitute the silly phrase. <strong>P<\/strong>lease <strong>E<\/strong>xcuse <strong>M<\/strong>y <strong>D<\/strong>ear <strong>A<\/strong>unt <strong>S<\/strong>ally.\r\n<table id=\"fs-id1786633\" class=\"unnumbered\" summary=\"This table has five rows and two columns. The first row spans both columns and is a header reading \">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th colspan=\"2\"><strong>Order of Operations<\/strong><\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td><strong>P<\/strong>lease<\/td>\r\n<td><strong>P<\/strong>arentheses<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td><strong>E<\/strong>xcuse<\/td>\r\n<td><strong>E<\/strong>xponents<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td><strong>M<\/strong>y <strong>D<\/strong>ear<\/td>\r\n<td><strong>M<\/strong>ultiplication and <strong>D<\/strong>ivision<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td><strong>A<\/strong>unt <strong>S<\/strong>ally<\/td>\r\n<td><strong>A<\/strong>ddition and <strong>S<\/strong>ubtraction<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIt\u2019s good that \u2018<strong>M<\/strong>y <strong>D<\/strong>ear\u2019 goes together, as this reminds us that <strong>m<\/strong>ultiplication and <strong>d<\/strong>ivision have equal priority. We do not always do multiplication before division or always do division before multiplication. We do them in order from left to right.\r\nSimilarly, \u2018<strong>A<\/strong>unt <strong>S<\/strong>ally\u2019 goes together and so reminds us that <strong>a<\/strong>ddition and <strong>s<\/strong>ubtraction also have equal priority and we do them in order from left to right.\r\n\r\n&nbsp;\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<td>1.<\/td>\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 <strong>p<\/strong>arentheses? No.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Are there any <strong>e<\/strong>xponents? No.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Is there any <strong>m<\/strong>ultiplication or <strong>d<\/strong>ivision? 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<td>2.<\/td>\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 <strong>p<\/strong>arentheses? 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 <strong>e<\/strong>xponents? No.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Is there any <strong>m<\/strong>ultiplication or <strong>d<\/strong>ivision? 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&nbsp;\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<td>1.<\/td>\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 <strong>p<\/strong>arentheses? No.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Are there any <strong>e<\/strong>xponents? No.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Is there any <strong>m<\/strong>ultiplication or <strong>d<\/strong>ivision? 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<td>2.<\/td>\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 <strong>p<\/strong>arentheses? No.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Are there any <strong>e<\/strong>xponents? No.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Is there any <strong>m<\/strong>ultiplication or <strong>d<\/strong>ivision? 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&nbsp;\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\n&nbsp;\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<td><\/td>\r\n<td>[latex]18\\div 6+4(5-2)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Parentheses? Yes, subtract first.<\/td>\r\n<td>[latex]18\\div 6+4(\\color{red}{3})[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Exponents? No.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiplication or division? Yes.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Divide first because we multiply and divide left to right.<\/td>\r\n<td>[latex]\\color{red}{3}+4(3)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Any other multiplication or division? Yes.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]3+\\color{red}{12}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Any other multiplication or division? No.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Any addition or subtraction? Yes.<\/td>\r\n<td>[latex]15[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\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[reveal-answer q=\"221697\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"221697\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1164754129434\" class=\"unnumbered unstyled\" summary=\"The image shows the expression five plus two cubed plus three, open bracket, six minus three, open parentheses, four minus two close parentheses, close bracket. Are there any parentheses (or other grouping symbols)? Yes, there are parentheses within brackets. Start with the innermost grouping, the parentheses. Inside the parentheses subtract two from four to get two. The expression becomes five plus two cubed plus three, open bracket, six minus three, in parentheses, two, close bracket. Within the brackets is six minus three, in parentheses, two. Multiply three by two since multiplication comes before subtraction. The expression becomes five plus two cubed plus three, in brackets, six minus six. Finish simplifying inside the brackets by performing the subtraction. Six minus six leaves zero. The expression is now five plus two cubed plus three, in brackets, zero. Are there exponents? Yes, two cubed is eight and the expression becomes five plus eight plus three, in brackets, zero. Is there any multiplication or division? Yes, multiply three by zero to get zero. The expression is now five plus eight plus zero. The remaining operations are both addition. Perform the addition from left to right. five plus eight is thirteen. Thirteen plus zero is thirteen.\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]5+{2}^{3}+ 3[6-3(4-2)][\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Are there any parentheses (or other grouping symbol)? Yes.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Focus on the parentheses that are inside the brackets.<\/td>\r\n<td>[latex]5+{2}^{3}+ 3[6-3\\color{red}{(4-2)}][\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract.<\/td>\r\n<td>[latex]5+{2}^{3}+3[6-\\color{red}{3(2)}][\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Continue inside the brackets and multiply.<\/td>\r\n<td>[latex]5+{2}^{3}+3[6-\\color{red}{6}][\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Continue inside the brackets and subtract.<\/td>\r\n<td>[latex]5+{2}^{3}+3[\\color{red}{0}][\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The expression inside the brackets requires no further simplification.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Are there any exponents? Yes.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify exponents.<\/td>\r\n<td>[latex]5+\\color{red}{{2}^{3}}+3[0][\/latex]<\/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]5+8+\\color{red}{3[0]}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Is there any addition or subtraction? Yes.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add.<\/td>\r\n<td>[latex]\\color{red}{5+8}+0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add.<\/td>\r\n<td>[latex]\\color{red}{13+0}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]13[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\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\" 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>\r\n<td><\/td>\r\n<td>[latex]{2}^{3}+{3}^{4}\\div 3-{5}^{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>If an expression has several exponents, they may be simplified in the same step.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify exponents.<\/td>\r\n<td>[latex]\\color{red}{{2}^{3}}+\\color{red}{{3}^{4}}\\div 3-\\color{red}{{5}^{2}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Divide.<\/td>\r\n<td>[latex]8+\\color{red}{81\\div 3}-25[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add.<\/td>\r\n<td>[latex]\\color{red}{8+27}-25[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract.<\/td>\r\n<td>[latex]\\color{red}{35-25}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]10[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\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>\r\n&nbsp;","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Use the order of operations to simplify mathematical expressions<\/li>\n<li>Simplify mathematical expressions involving addition, subtraction, multiplication, division, and exponents<\/li>\n<\/ul>\n<\/div>\n<p>&nbsp;<\/p>\n<h2>Simplify Expressions Using the Order of Operations<\/h2>\n<p>We\u2019ve introduced most of the symbols and notation used in algebra, but now we need to clarify the order of operations. 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. <strong>P<\/strong>arentheses and other Grouping Symbols<\/p>\n<ul id=\"fs-id1171104029952\">\n<li>Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first.<\/li>\n<\/ul>\n<p>2. <strong>E<\/strong>xponents<\/p>\n<ul id=\"fs-id1171104407077\">\n<li>Simplify all expressions with exponents.<\/li>\n<\/ul>\n<p>3. <strong>M<\/strong>ultiplication and <strong>D<\/strong>ivision<\/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. <strong>A<\/strong>ddition and <strong>S<\/strong>ubtraction<\/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<p>&nbsp;<\/p>\n<p>Students often ask, &#8220;How will I remember the order?&#8221; Here is a way to help you remember: Take the first letter of each key word and substitute the silly phrase. <strong>P<\/strong>lease <strong>E<\/strong>xcuse <strong>M<\/strong>y <strong>D<\/strong>ear <strong>A<\/strong>unt <strong>S<\/strong>ally.<\/p>\n<table id=\"fs-id1786633\" class=\"unnumbered\" summary=\"This table has five rows and two columns. The first row spans both columns and is a header reading\">\n<thead>\n<tr valign=\"top\">\n<th colspan=\"2\"><strong>Order of Operations<\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td><strong>P<\/strong>lease<\/td>\n<td><strong>P<\/strong>arentheses<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><strong>E<\/strong>xcuse<\/td>\n<td><strong>E<\/strong>xponents<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><strong>M<\/strong>y <strong>D<\/strong>ear<\/td>\n<td><strong>M<\/strong>ultiplication and <strong>D<\/strong>ivision<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><strong>A<\/strong>unt <strong>S<\/strong>ally<\/td>\n<td><strong>A<\/strong>ddition and <strong>S<\/strong>ubtraction<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>It\u2019s good that \u2018<strong>M<\/strong>y <strong>D<\/strong>ear\u2019 goes together, as this reminds us that <strong>m<\/strong>ultiplication and <strong>d<\/strong>ivision have equal priority. We do not always do multiplication before division or always do division before multiplication. We do them in order from left to right.<br \/>\nSimilarly, \u2018<strong>A<\/strong>unt <strong>S<\/strong>ally\u2019 goes together and so reminds us that <strong>a<\/strong>ddition and <strong>s<\/strong>ubtraction also have equal priority and we do them in order from left to right.<\/p>\n<p>&nbsp;<\/p>\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<td>1.<\/td>\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 <strong>p<\/strong>arentheses? No.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Are there any <strong>e<\/strong>xponents? No.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Is there any <strong>m<\/strong>ultiplication or <strong>d<\/strong>ivision? 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<td>2.<\/td>\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 <strong>p<\/strong>arentheses? 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 <strong>e<\/strong>xponents? No.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Is there any <strong>m<\/strong>ultiplication or <strong>d<\/strong>ivision? 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<p>&nbsp;<\/p>\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<td>1.<\/td>\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 <strong>p<\/strong>arentheses? No.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Are there any <strong>e<\/strong>xponents? No.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Is there any <strong>m<\/strong>ultiplication or <strong>d<\/strong>ivision? 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<td>2.<\/td>\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 <strong>p<\/strong>arentheses? No.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Are there any <strong>e<\/strong>xponents? No.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Is there any <strong>m<\/strong>ultiplication or <strong>d<\/strong>ivision? 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<p>&nbsp;<\/p>\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>&nbsp;<\/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<td><\/td>\n<td>[latex]18\\div 6+4(5-2)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Parentheses? Yes, subtract first.<\/td>\n<td>[latex]18\\div 6+4(\\color{red}{3})[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Exponents? No.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Multiplication or division? Yes.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Divide first because we multiply and divide left to right.<\/td>\n<td>[latex]\\color{red}{3}+4(3)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Any other multiplication or division? Yes.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]3+\\color{red}{12}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Any other multiplication or division? No.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Any addition or subtraction? Yes.<\/td>\n<td>[latex]15[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\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<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q221697\">Show Solution<\/span><\/p>\n<div id=\"q221697\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1164754129434\" class=\"unnumbered unstyled\" summary=\"The image shows the expression five plus two cubed plus three, open bracket, six minus three, open parentheses, four minus two close parentheses, close bracket. Are there any parentheses (or other grouping symbols)? Yes, there are parentheses within brackets. Start with the innermost grouping, the parentheses. Inside the parentheses subtract two from four to get two. The expression becomes five plus two cubed plus three, open bracket, six minus three, in parentheses, two, close bracket. Within the brackets is six minus three, in parentheses, two. Multiply three by two since multiplication comes before subtraction. The expression becomes five plus two cubed plus three, in brackets, six minus six. Finish simplifying inside the brackets by performing the subtraction. Six minus six leaves zero. The expression is now five plus two cubed plus three, in brackets, zero. Are there exponents? Yes, two cubed is eight and the expression becomes five plus eight plus three, in brackets, zero. Is there any multiplication or division? Yes, multiply three by zero to get zero. The expression is now five plus eight plus zero. The remaining operations are both addition. Perform the addition from left to right. five plus eight is thirteen. Thirteen plus zero is thirteen.\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]5+{2}^{3}+ 3[6-3(4-2)][\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Are there any parentheses (or other grouping symbol)? Yes.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Focus on the parentheses that are inside the brackets.<\/td>\n<td>[latex]5+{2}^{3}+ 3[6-3\\color{red}{(4-2)}][\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract.<\/td>\n<td>[latex]5+{2}^{3}+3[6-\\color{red}{3(2)}][\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Continue inside the brackets and multiply.<\/td>\n<td>[latex]5+{2}^{3}+3[6-\\color{red}{6}][\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Continue inside the brackets and subtract.<\/td>\n<td>[latex]5+{2}^{3}+3[\\color{red}{0}][\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The expression inside the brackets requires no further simplification.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Are there any exponents? Yes.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Simplify exponents.<\/td>\n<td>[latex]5+\\color{red}{{2}^{3}}+3[0][\/latex]<\/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]5+8+\\color{red}{3[0]}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Is there any addition or subtraction? Yes.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Add.<\/td>\n<td>[latex]\\color{red}{5+8}+0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add.<\/td>\n<td>[latex]\\color{red}{13+0}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]13[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\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\" 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>\n<td><\/td>\n<td>[latex]{2}^{3}+{3}^{4}\\div 3-{5}^{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>If an expression has several exponents, they may be simplified in the same step.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Simplify exponents.<\/td>\n<td>[latex]\\color{red}{{2}^{3}}+\\color{red}{{3}^{4}}\\div 3-\\color{red}{{5}^{2}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Divide.<\/td>\n<td>[latex]8+\\color{red}{81\\div 3}-25[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add.<\/td>\n<td>[latex]\\color{red}{8+27}-25[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract.<\/td>\n<td>[latex]\\color{red}{35-25}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]10[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\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<p>&nbsp;<\/p>\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-9207\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Ex: Evaluate an Expression Using the Order of Operations. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/qFUvF5-w9o0\">https:\/\/youtu.be\/qFUvF5-w9o0<\/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>Example 3: Evaluate An Expression Using The Order of Operation. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/8b-rf2AW3Ac\">https:\/\/youtu.be\/8b-rf2AW3Ac<\/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>Question ID: 144748, 144751, 144756, 144758, 144759, 144762. <strong>Authored by<\/strong>: Alyson Day. <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, Specific attribution<\/div><ul class=\"citation-list\"><li>Prealgebra. <strong>Provided by<\/strong>: OpenStax. <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>: Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757<\/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":17533,"menu_order":5,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Ex: Evaluate an Expression Using the Order of Operations\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/qFUvF5-w9o0\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Example 3: Evaluate An Expression Using The Order of Operation\",\"author\":\"James Sousa 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