{"id":3678,"date":"2020-01-29T02:30:58","date_gmt":"2020-01-29T02:30:58","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/mathforlibscoreq\/?post_type=chapter&#038;p=3678"},"modified":"2021-02-05T23:51:02","modified_gmt":"2021-02-05T23:51:02","slug":"evaluating-and-simplifying-expressions-using-the-commutative-and-associative-properties","status":"web-only","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/chapter\/evaluating-and-simplifying-expressions-using-the-commutative-and-associative-properties\/","title":{"raw":"Evaluating and Simplifying Expressions Using the Commutative and Associative Properties","rendered":"Evaluating and Simplifying Expressions Using the Commutative and Associative Properties"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Evaluate algebraic expressions for a given value using the commutative and associative properties of addition and multiplication<\/li>\r\n \t<li>Simplify algebraic expressions\u00a0using the commutative and associative properties of addition and multiplication<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Evaluate Expressions using the Commutative and Associative Properties<\/h2>\r\nThe commutative and associative properties can make it easier to evaluate some algebraic expressions. Since order does not matter when adding or multiplying three or more terms, we can rearrange and re-group terms to make our work easier, as the next several examples illustrate.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nEvaluate each expression when [latex]x=\\Large\\frac{7}{8}[\/latex].\r\n<ol id=\"eip-id1168467300574\" class=\"circled\">\r\n \t<li>[latex]x+0.37+\\left(-x\\right)[\/latex]<\/li>\r\n \t<li>[latex]x+\\left(-x\\right)+0.37[\/latex]<\/li>\r\n<\/ol>\r\nSolution:\r\n<table id=\"eip-id1168469782831\" class=\"unnumbered unstyled\" style=\"width: 85%\" summary=\"The image shows the expression x plus 0.37 plus negative x. Substitute the fraction 7 eights for x and the expression becomes 7 eights plus 0.37 plus negative 7 eighths. Convert the fractions to decimals to get 0.875 plus 0.37 plus negative 0.875. Adding from left to right 0.875 plus 0.37 becomes 1.245 and the expression becomes 1.245 plus negative 0.875. Adding a negative is the same as subtraction so now the expression can be written as 1.245 minus 0.875. Perform the subtraction to get 0.37.\">\r\n<tbody>\r\n<tr style=\"height: 15.5469px\">\r\n<td style=\"height: 15.5469px\">1.<\/td>\r\n<td style=\"height: 15.5469px\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px\">\r\n<td style=\"height: 15px\"><\/td>\r\n<td style=\"height: 15px\">[latex]x+0.37+(--x)[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px\">\r\n<td style=\"height: 15px\">Substitute [latex]\\Large\\frac{7}{8}[\/latex] for [latex]x[\/latex] .<\/td>\r\n<td style=\"height: 15px\">[latex]\\color{red}{\\Large\\frac{7}{8}}\\normalsize +0.37+(--\\color{red}{\\Large\\frac{7}{8}})[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px\">\r\n<td style=\"height: 15px\">Convert fractions to decimals.<\/td>\r\n<td style=\"height: 15px\">[latex]0.875+0.37+(--0.875)[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px\">\r\n<td style=\"height: 15px\">Add left to right.<\/td>\r\n<td style=\"height: 15px\">[latex]1.245--0.875[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px\">\r\n<td style=\"height: 15px\">Subtract.<\/td>\r\n<td style=\"height: 15px\">[latex]0.37[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168468470071\" class=\"unnumbered unstyled\" style=\"width: 85%\" summary=\"The image shows the expression x plus negative x plus 0.37. Substitute the fraction 7 eights for x and the expression becomes 7 eights plus negative 7 eighths plus 0.37. Add the opposites first so that only 0.37 is left.\">\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]x+(--x)+0.37[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute [latex]\\Large\\frac{7}{8}[\/latex] for x.<\/td>\r\n<td>[latex]\\color{red}{\\Large\\frac{7}{8}}\\normalsize +(--\\color{red}{\\Large\\frac{7}{8}})+0.37[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add opposites first.<\/td>\r\n<td>[latex]0.37[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWhat was the difference between part 1 and part 2? Only the order changed. By the Commutative Property of Addition, [latex]x+0.37+\\left(-x\\right)=x+\\left(-x\\right)+0.37[\/latex]. But wasn\u2019t part 2 much easier?\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]145792[\/ohm_question]\r\n\r\n<\/div>\r\nLet\u2019s do one more, this time with multiplication.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nEvaluate each expression when [latex]n=17[\/latex].\r\n1. [latex]\\Large\\frac{4}{3}\\left(\\Large\\frac{3}{4}\\normalsize n\\Large\\right)[\/latex]\r\n2. [latex]\\left(\\Large\\frac{4}{3}\\normalsize\\cdot\\Large\\frac{3}{4}\\right)\\normalsize n[\/latex]\r\n[reveal-answer q=\"889127\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"889127\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168469752671\" class=\"unnumbered unstyled\" style=\"width: 85%\" summary=\"The image shows the expression 4 thirds times the quantity 3 quarters n in parentheses. Substitute 17 for n and the expression becomes 4 thirds times the quantity 3 quarters times 17 in parentheses. Simplify inside the parentheses first. Multiply 3 quarters by 17 to get 51 quarters. The expression becomes 4 thirds times the quantity 51 quarters in parentheses. Multiply to get 17.\">\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]\\Large\\frac{4}{3}(\\frac{3}{4}\\normalsize n\\Large)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute 17 for n.<\/td>\r\n<td>[latex]\\Large\\frac{4}{3}(\\frac{3}{4}\\normalsize\\cdot\\color{red}{17}\\Large)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply in the parentheses first.<\/td>\r\n<td>[latex]\\Large\\frac{4}{3}(\\frac{51}{4})[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply again.<\/td>\r\n<td>[latex]17[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168468293112\" class=\"unnumbered unstyled\" style=\"width: 85%\" summary=\"The image shows the expression the quantity 4 thirds times 3 quarters in parentheses times n. Substitute 17 for n. Multiply inside the parentheses first to get 1. The equation is 1 times 17 to get 17.\">\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]\\Large(\\frac{4}{3}\\normalsize\\cdot\\Large\\frac{3}{4})\\normalsize n[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute 17 for n.<\/td>\r\n<td>[latex]\\Large(\\frac{4}{3}\\normalsize\\cdot\\Large\\frac{3}{4})\\normalsize\\cdot\\color{red}{17}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply. The product of reciprocals is 1.<\/td>\r\n<td>[latex](1)\\cdot17[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply again.<\/td>\r\n<td>[latex]17[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWhat was the difference between part 1 and part 2 here? Only the grouping changed. By the Associative Property of Multiplication, [latex]\\Large\\frac{4}{3}\\left(\\Large\\frac{3}{4}\\normalsize n\\Large\\right)\\normalsize =\\Large\\left(\\frac{4}{3}\\normalsize\\cdot\\Large\\frac{3}{4}\\right)\\normalsize n[\/latex]. By carefully choosing how to group the factors, we can make the work easier.\r\n\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]145796[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Simplify Expressions Using the Commutative and Associative Properties<\/h2>\r\nWhen we have to simplify algebraic expressions, we can often make the work easier by applying the Commutative or Associative Property first instead of automatically following the order of operations. Notice that in the first example,\u00a0part 2 was easier to simplify than part 1 because the opposites were next to each other and their sum is [latex]0[\/latex]. Likewise, part 2 in the second example was easier, with the reciprocals grouped together, because their product is [latex]1[\/latex]. In the next few examples, we\u2019ll use our number sense to look for ways to apply these properties to make our work easier.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]-84n+\\left(-73n\\right)+84n[\/latex]\r\n[reveal-answer q=\"679824\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"679824\"]\r\n\r\nSolution:\r\nNotice the first and third terms are opposites, so we can use the commutative property of addition to reorder the terms.\r\n<table id=\"eip-id1168469525330\" class=\"unnumbered unstyled\" style=\"width: 85%\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-84n+\\left(-73n\\right)+84n[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Re-order the terms.<\/td>\r\n<td>[latex]-84n+84n+\\left(-73n\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add left to right.<\/td>\r\n<td>[latex]0+\\left(-73n\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add.<\/td>\r\n<td>[latex]-73n[\/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]145797[\/ohm_question]\r\n\r\n<\/div>\r\nWatch the following video for more similar examples of how to use the associative and commutative properties to simplify expressions.\r\n\r\nhttps:\/\/youtu.be\/8vMRywgaqOE\r\n\r\nNow we will see how recognizing reciprocals is helpful. Before multiplying left to right, look for reciprocals\u2014their product is [latex]1[\/latex].\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]\\Large\\frac{7}{15}\\cdot \\frac{8}{23}\\cdot \\frac{15}{7}[\/latex]\r\n[reveal-answer q=\"543511\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"543511\"]\r\n\r\nSolution:\r\nNotice the first and third terms are reciprocals, so we can use the Commutative Property of Multiplication to reorder the factors.\r\n<table id=\"eip-id1168466660267\" class=\"unnumbered unstyled\" style=\"width: 85%\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\Large\\frac{7}{15}\\cdot \\frac{8}{23}\\cdot \\frac{15}{7}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Re-order the terms.<\/td>\r\n<td>[latex]\\Large\\frac{7}{15}\\cdot \\frac{15}{7}\\cdot \\frac{8}{23}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply left to right.<\/td>\r\n<td>[latex]1\\cdot\\Large\\frac{8}{23}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]\\Large\\frac{8}{23}[\/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]145798[\/ohm_question]\r\n\r\n<\/div>\r\nIn expressions where we need to add or subtract three or more fractions, combine those with a common denominator first.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]\\Large\\left(\\frac{5}{13}\\normalsize +\\Large\\frac{3}{4}\\right)\\normalsize +\\Large\\frac{1}{4}[\/latex]\r\n[reveal-answer q=\"414059\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"414059\"]\r\n\r\nSolution:\r\nNotice that the second and third terms have a common denominator, so this work will be easier if we change the grouping.\r\n<table id=\"eip-id1168469835510\" class=\"unnumbered unstyled\" style=\"width: 85%\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\Large\\left(\\frac{5}{13}\\normalsize +\\Large\\frac{3}{4}\\right)\\normalsize +\\Large\\frac{1}{4}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Group the terms with a common denominator.<\/td>\r\n<td>[latex]\\Large\\frac{5}{13}\\normalsize +\\Large\\left(\\frac{3}{4}\\normalsize +\\Large\\frac{1}{4}\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add in the parentheses first.<\/td>\r\n<td>[latex]\\Large\\frac{5}{13}\\normalsize +\\Large\\left(\\frac{4}{4}\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify the fraction.<\/td>\r\n<td>[latex]\\Large\\frac{5}{13}\\normalsize +1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add.<\/td>\r\n<td>[latex]1\\Large\\frac{5}{13}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Convert to an improper fraction.<\/td>\r\n<td>[latex]\\Large\\frac{18}{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]145799[\/ohm_question]\r\n\r\n<\/div>\r\nWhen adding and subtracting three or more terms involving decimals, look for terms that combine to give whole numbers.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]\\left(6.47q+9.99q\\right)+1.01q[\/latex]\r\n[reveal-answer q=\"964224\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"964224\"]\r\n\r\nSolution:\r\nNotice that the sum of the second and third coefficients is a whole number.\r\n<table id=\"eip-id1168468331350\" class=\"unnumbered unstyled\" style=\"width: 85%\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\left(6.47q+9.99q\\right)+1.01q[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Change the grouping.<\/td>\r\n<td>[latex]6.47q+\\left(9.99q+1.01q\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add in the parentheses first.<\/td>\r\n<td>[latex]6.47q+\\left(11.00q\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add.<\/td>\r\n<td>[latex]17.47q[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nMany people have good number sense when they deal with money. Think about adding [latex]99[\/latex] cents and [latex]1[\/latex] cent. Do you see how this applies to adding [latex]9.99+1.01?[\/latex]\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]145800[\/ohm_question]\r\n\r\n<\/div>\r\nWhen simplifying expressions that contain variables, we can use the commutative and associative properties to re-order or regroup terms, as shown in the next pair of examples.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]6\\left(9x\\right)[\/latex]\r\n[reveal-answer q=\"587700\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"587700\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168467313641\" class=\"unnumbered unstyled\" style=\"width: 85%\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]6\\left(9x\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the associative property of multiplication to re-group.<\/td>\r\n<td>[latex]\\left(6\\cdot 9\\right)x[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply in the parentheses.<\/td>\r\n<td>[latex]54x[\/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]145973[\/ohm_question]\r\n\r\n<\/div>\r\nIn The Language of Algebra, we learned to combine like terms by rearranging an expression so the like terms were together. We simplified the expression [latex]3x+7+4x+5[\/latex] by rewriting it as [latex]3x+4x+7+5[\/latex] and then simplified it to [latex]7x+12[\/latex]. We were using the Commutative Property of Addition.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]18p+6q+\\left(-15p\\right)+5q[\/latex]\r\n[reveal-answer q=\"506787\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"506787\"]\r\n\r\nSolution:\r\nUse the Commutative Property of Addition to re-order so that like terms are together.\r\n<table id=\"eip-id1168466129786\" class=\"unnumbered unstyled\" style=\"width: 85%\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]18p+6q+\\left(-15p\\right)+5q[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Re-order terms.<\/td>\r\n<td>[latex]18p+\\left(-15p\\right)+6q+5q[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Combine like terms.<\/td>\r\n<td>[latex]3p+11q[\/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\nSimplify: [latex]23r+14s+9r+\\left(-15s\\right)[\/latex]\r\n\r\n[reveal-answer q=\"30457\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"30457\"]\r\n\r\n[latex]32r\u2212s[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\nSimplify: [latex]37m+21n+4m+\\left(-15n\\right)[\/latex]\r\n\r\n[reveal-answer q=\"87324\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"87324\"]\r\n\r\n[latex]41m+6n[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Evaluate algebraic expressions for a given value using the commutative and associative properties of addition and multiplication<\/li>\n<li>Simplify algebraic expressions\u00a0using the commutative and associative properties of addition and multiplication<\/li>\n<\/ul>\n<\/div>\n<h2>Evaluate Expressions using the Commutative and Associative Properties<\/h2>\n<p>The commutative and associative properties can make it easier to evaluate some algebraic expressions. Since order does not matter when adding or multiplying three or more terms, we can rearrange and re-group terms to make our work easier, as the next several examples illustrate.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Evaluate each expression when [latex]x=\\Large\\frac{7}{8}[\/latex].<\/p>\n<ol id=\"eip-id1168467300574\" class=\"circled\">\n<li>[latex]x+0.37+\\left(-x\\right)[\/latex]<\/li>\n<li>[latex]x+\\left(-x\\right)+0.37[\/latex]<\/li>\n<\/ol>\n<p>Solution:<\/p>\n<table id=\"eip-id1168469782831\" class=\"unnumbered unstyled\" style=\"width: 85%\" summary=\"The image shows the expression x plus 0.37 plus negative x. Substitute the fraction 7 eights for x and the expression becomes 7 eights plus 0.37 plus negative 7 eighths. Convert the fractions to decimals to get 0.875 plus 0.37 plus negative 0.875. Adding from left to right 0.875 plus 0.37 becomes 1.245 and the expression becomes 1.245 plus negative 0.875. Adding a negative is the same as subtraction so now the expression can be written as 1.245 minus 0.875. Perform the subtraction to get 0.37.\">\n<tbody>\n<tr style=\"height: 15.5469px\">\n<td style=\"height: 15.5469px\">1.<\/td>\n<td style=\"height: 15.5469px\"><\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"height: 15px\"><\/td>\n<td style=\"height: 15px\">[latex]x+0.37+(--x)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"height: 15px\">Substitute [latex]\\Large\\frac{7}{8}[\/latex] for [latex]x[\/latex] .<\/td>\n<td style=\"height: 15px\">[latex]\\color{red}{\\Large\\frac{7}{8}}\\normalsize +0.37+(--\\color{red}{\\Large\\frac{7}{8}})[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"height: 15px\">Convert fractions to decimals.<\/td>\n<td style=\"height: 15px\">[latex]0.875+0.37+(--0.875)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"height: 15px\">Add left to right.<\/td>\n<td style=\"height: 15px\">[latex]1.245--0.875[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"height: 15px\">Subtract.<\/td>\n<td style=\"height: 15px\">[latex]0.37[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168468470071\" class=\"unnumbered unstyled\" style=\"width: 85%\" summary=\"The image shows the expression x plus negative x plus 0.37. Substitute the fraction 7 eights for x and the expression becomes 7 eights plus negative 7 eighths plus 0.37. Add the opposites first so that only 0.37 is left.\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]x+(--x)+0.37[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]\\Large\\frac{7}{8}[\/latex] for x.<\/td>\n<td>[latex]\\color{red}{\\Large\\frac{7}{8}}\\normalsize +(--\\color{red}{\\Large\\frac{7}{8}})+0.37[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add opposites first.<\/td>\n<td>[latex]0.37[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>What was the difference between part 1 and part 2? Only the order changed. By the Commutative Property of Addition, [latex]x+0.37+\\left(-x\\right)=x+\\left(-x\\right)+0.37[\/latex]. But wasn\u2019t part 2 much easier?<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm145792\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145792&theme=oea&iframe_resize_id=ohm145792&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Let\u2019s do one more, this time with multiplication.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Evaluate each expression when [latex]n=17[\/latex].<br \/>\n1. [latex]\\Large\\frac{4}{3}\\left(\\Large\\frac{3}{4}\\normalsize n\\Large\\right)[\/latex]<br \/>\n2. [latex]\\left(\\Large\\frac{4}{3}\\normalsize\\cdot\\Large\\frac{3}{4}\\right)\\normalsize n[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q889127\">Show Solution<\/span><\/p>\n<div id=\"q889127\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168469752671\" class=\"unnumbered unstyled\" style=\"width: 85%\" summary=\"The image shows the expression 4 thirds times the quantity 3 quarters n in parentheses. Substitute 17 for n and the expression becomes 4 thirds times the quantity 3 quarters times 17 in parentheses. Simplify inside the parentheses first. Multiply 3 quarters by 17 to get 51 quarters. The expression becomes 4 thirds times the quantity 51 quarters in parentheses. Multiply to get 17.\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]\\Large\\frac{4}{3}(\\frac{3}{4}\\normalsize n\\Large)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute 17 for n.<\/td>\n<td>[latex]\\Large\\frac{4}{3}(\\frac{3}{4}\\normalsize\\cdot\\color{red}{17}\\Large)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply in the parentheses first.<\/td>\n<td>[latex]\\Large\\frac{4}{3}(\\frac{51}{4})[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply again.<\/td>\n<td>[latex]17[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168468293112\" class=\"unnumbered unstyled\" style=\"width: 85%\" summary=\"The image shows the expression the quantity 4 thirds times 3 quarters in parentheses times n. Substitute 17 for n. Multiply inside the parentheses first to get 1. The equation is 1 times 17 to get 17.\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]\\Large(\\frac{4}{3}\\normalsize\\cdot\\Large\\frac{3}{4})\\normalsize n[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute 17 for n.<\/td>\n<td>[latex]\\Large(\\frac{4}{3}\\normalsize\\cdot\\Large\\frac{3}{4})\\normalsize\\cdot\\color{red}{17}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply. The product of reciprocals is 1.<\/td>\n<td>[latex](1)\\cdot17[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply again.<\/td>\n<td>[latex]17[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>What was the difference between part 1 and part 2 here? Only the grouping changed. By the Associative Property of Multiplication, [latex]\\Large\\frac{4}{3}\\left(\\Large\\frac{3}{4}\\normalsize n\\Large\\right)\\normalsize =\\Large\\left(\\frac{4}{3}\\normalsize\\cdot\\Large\\frac{3}{4}\\right)\\normalsize n[\/latex]. By carefully choosing how to group the factors, we can make the work easier.<\/p>\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=\"ohm145796\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145796&theme=oea&iframe_resize_id=ohm145796&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Simplify Expressions Using the Commutative and Associative Properties<\/h2>\n<p>When we have to simplify algebraic expressions, we can often make the work easier by applying the Commutative or Associative Property first instead of automatically following the order of operations. Notice that in the first example,\u00a0part 2 was easier to simplify than part 1 because the opposites were next to each other and their sum is [latex]0[\/latex]. Likewise, part 2 in the second example was easier, with the reciprocals grouped together, because their product is [latex]1[\/latex]. In the next few examples, we\u2019ll use our number sense to look for ways to apply these properties to make our work easier.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]-84n+\\left(-73n\\right)+84n[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q679824\">Show Solution<\/span><\/p>\n<div id=\"q679824\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<br \/>\nNotice the first and third terms are opposites, so we can use the commutative property of addition to reorder the terms.<\/p>\n<table id=\"eip-id1168469525330\" class=\"unnumbered unstyled\" style=\"width: 85%\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]-84n+\\left(-73n\\right)+84n[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Re-order the terms.<\/td>\n<td>[latex]-84n+84n+\\left(-73n\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add left to right.<\/td>\n<td>[latex]0+\\left(-73n\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add.<\/td>\n<td>[latex]-73n[\/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=\"ohm145797\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145797&theme=oea&iframe_resize_id=ohm145797&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Watch the following video for more similar examples of how to use the associative and commutative properties to simplify expressions.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Review Properties of Real Numbers While Simplifying Expressions\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/8vMRywgaqOE?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>Now we will see how recognizing reciprocals is helpful. Before multiplying left to right, look for reciprocals\u2014their product is [latex]1[\/latex].<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]\\Large\\frac{7}{15}\\cdot \\frac{8}{23}\\cdot \\frac{15}{7}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q543511\">Show Solution<\/span><\/p>\n<div id=\"q543511\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<br \/>\nNotice the first and third terms are reciprocals, so we can use the Commutative Property of Multiplication to reorder the factors.<\/p>\n<table id=\"eip-id1168466660267\" class=\"unnumbered unstyled\" style=\"width: 85%\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]\\Large\\frac{7}{15}\\cdot \\frac{8}{23}\\cdot \\frac{15}{7}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Re-order the terms.<\/td>\n<td>[latex]\\Large\\frac{7}{15}\\cdot \\frac{15}{7}\\cdot \\frac{8}{23}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply left to right.<\/td>\n<td>[latex]1\\cdot\\Large\\frac{8}{23}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]\\Large\\frac{8}{23}[\/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=\"ohm145798\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145798&theme=oea&iframe_resize_id=ohm145798&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In expressions where we need to add or subtract three or more fractions, combine those with a common denominator first.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]\\Large\\left(\\frac{5}{13}\\normalsize +\\Large\\frac{3}{4}\\right)\\normalsize +\\Large\\frac{1}{4}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q414059\">Show Solution<\/span><\/p>\n<div id=\"q414059\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<br \/>\nNotice that the second and third terms have a common denominator, so this work will be easier if we change the grouping.<\/p>\n<table id=\"eip-id1168469835510\" class=\"unnumbered unstyled\" style=\"width: 85%\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]\\Large\\left(\\frac{5}{13}\\normalsize +\\Large\\frac{3}{4}\\right)\\normalsize +\\Large\\frac{1}{4}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Group the terms with a common denominator.<\/td>\n<td>[latex]\\Large\\frac{5}{13}\\normalsize +\\Large\\left(\\frac{3}{4}\\normalsize +\\Large\\frac{1}{4}\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add in the parentheses first.<\/td>\n<td>[latex]\\Large\\frac{5}{13}\\normalsize +\\Large\\left(\\frac{4}{4}\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify the fraction.<\/td>\n<td>[latex]\\Large\\frac{5}{13}\\normalsize +1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add.<\/td>\n<td>[latex]1\\Large\\frac{5}{13}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Convert to an improper fraction.<\/td>\n<td>[latex]\\Large\\frac{18}{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=\"ohm145799\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145799&theme=oea&iframe_resize_id=ohm145799&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>When adding and subtracting three or more terms involving decimals, look for terms that combine to give whole numbers.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]\\left(6.47q+9.99q\\right)+1.01q[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q964224\">Show Solution<\/span><\/p>\n<div id=\"q964224\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<br \/>\nNotice that the sum of the second and third coefficients is a whole number.<\/p>\n<table id=\"eip-id1168468331350\" class=\"unnumbered unstyled\" style=\"width: 85%\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]\\left(6.47q+9.99q\\right)+1.01q[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Change the grouping.<\/td>\n<td>[latex]6.47q+\\left(9.99q+1.01q\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add in the parentheses first.<\/td>\n<td>[latex]6.47q+\\left(11.00q\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add.<\/td>\n<td>[latex]17.47q[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>Many people have good number sense when they deal with money. Think about adding [latex]99[\/latex] cents and [latex]1[\/latex] cent. Do you see how this applies to adding [latex]9.99+1.01?[\/latex]<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm145800\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145800&theme=oea&iframe_resize_id=ohm145800&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>When simplifying expressions that contain variables, we can use the commutative and associative properties to re-order or regroup terms, as shown in the next pair of examples.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]6\\left(9x\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q587700\">Show Solution<\/span><\/p>\n<div id=\"q587700\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168467313641\" class=\"unnumbered unstyled\" style=\"width: 85%\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]6\\left(9x\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the associative property of multiplication to re-group.<\/td>\n<td>[latex]\\left(6\\cdot 9\\right)x[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply in the parentheses.<\/td>\n<td>[latex]54x[\/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=\"ohm145973\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145973&theme=oea&iframe_resize_id=ohm145973&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In The Language of Algebra, we learned to combine like terms by rearranging an expression so the like terms were together. We simplified the expression [latex]3x+7+4x+5[\/latex] by rewriting it as [latex]3x+4x+7+5[\/latex] and then simplified it to [latex]7x+12[\/latex]. We were using the Commutative Property of Addition.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]18p+6q+\\left(-15p\\right)+5q[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q506787\">Show Solution<\/span><\/p>\n<div id=\"q506787\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<br \/>\nUse the Commutative Property of Addition to re-order so that like terms are together.<\/p>\n<table id=\"eip-id1168466129786\" class=\"unnumbered unstyled\" style=\"width: 85%\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]18p+6q+\\left(-15p\\right)+5q[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Re-order terms.<\/td>\n<td>[latex]18p+\\left(-15p\\right)+6q+5q[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Combine like terms.<\/td>\n<td>[latex]3p+11q[\/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>Simplify: [latex]23r+14s+9r+\\left(-15s\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q30457\">Show Solution<\/span><\/p>\n<div id=\"q30457\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]32r\u2212s[\/latex]<\/p>\n<\/div>\n<\/div>\n<p>Simplify: [latex]37m+21n+4m+\\left(-15n\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q87324\">Show Solution<\/span><\/p>\n<div id=\"q87324\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]41m+6n[\/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-3678\">\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>Review Properties of Real Numbers While Simplifying Expressions. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/8vMRywgaqOE\">https:\/\/youtu.be\/8vMRywgaqOE<\/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 145792, 145796, 145797, 145798, 145799, 145973. <strong>Authored 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 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":25777,"menu_order":6,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Prealgebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757\"},{\"type\":\"original\",\"description\":\"Review Properties of Real Numbers While Simplifying Expressions\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen 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