7.3: Commutative and Associative Properties (Part 2)
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Simplify Expressions Using the Commutative and Associative Properties
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 Example 7.2.4 part (b) was easier to simplify than part (a) because the opposites were next to each other and their sum is 0. Likewise, part (b) in Example 7.2.5 was easier, with the reciprocals grouped together, because their product is 1. In the next few examples, we’ll use our number sense to look for ways to apply these properties to make our work easier.
Simplify: −84n + (−73n) + 84n.
Solution
Notice the first and third terms are opposites, so we can use the commutative property of addition to reorder the terms.
| Re-order the terms. | −84n + 84n + (−73n) |
| Add left to right. | 0 + (−73n) |
| Add. | −73n |
Simplify: −27a + (−48a) + 27a.
- Answer
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\(-48a\)
Simplify: 39x + (−92x) + (−39x).
- Answer
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\(-92x\)
Now we will see how recognizing reciprocals is helpful. Before multiplying left to right, look for reciprocals—their product is 1.
Simplify: \(\dfrac{7}{15} \cdot \dfrac{8}{23} \cdot \dfrac{15}{7}\).
Solution
Notice the first and third terms are reciprocals, so we can use the Commutative Property of Multiplication to reorder the factors.
| Re-order the terms. | $$\dfrac{7}{15} \cdot \dfrac{15}{7} \cdot \dfrac{8}{23}$$ |
| Multiply left to right. | $$1 \cdot \dfrac{8}{23}$$ |
| Multiply. | $$\dfrac{8}{23}$$ |
Simplify: \(\dfrac{9}{16} \cdot \dfrac{5}{49} \cdot \dfrac{16}{9}\).
- Answer
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\(\frac{5}{49}\)
Simplify: \(\dfrac{6}{17} \cdot \dfrac{11}{25} \cdot \dfrac{17}{6}\).
- Answer
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\(\frac{11}{25}\)
In expressions where we need to add or subtract three or more fractions, combine those with a common denominator first.
Simplify: \(\left(\dfrac{5}{13} + \dfrac{3}{4}\right) + \dfrac{1}{4}\).
Solution
Notice that the second and third terms have a common denominator, so this work will be easier if we change the grouping.
| Group the terms with a common denominator. | $$\dfrac{5}{13} + \left(\dfrac{3}{4} + \dfrac{1}{4}\right)$$ |
| Add in the parentheses first. | $$\dfrac{5}{13} + \left(\dfrac{4}{4}\right)$$ |
| Simplify the fraction. | $$\dfrac{5}{13} + 1$$ |
| Add. | $$1 \dfrac{5}{13}$$ |
| Convert to an improper fraction. | $$\dfrac{18}{13}$$ |
Simplify: \(\left(\dfrac{7}{15} + \dfrac{5}{8}\right) + \dfrac{3}{8}\).
- Answer
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\(\frac{22}{15}\)
Simplify: \(\left(\dfrac{2}{9} + \dfrac{7}{12}\right) + \dfrac{5}{12}\).
- Answer
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\(\frac{11}{9}\)
When adding and subtracting three or more terms involving decimals, look for terms that combine to give whole numbers.
Simplify: (6.47q + 9.99q) + 1.01q.
Solution
Notice that the sum of the second and third coefficients is a whole number.
| Change the grouping. | 6.47q + (9.99q + 1.01q) |
| Add in the parentheses first. | 6.47q + (11.00q) |
| Add. | 17.47q |
Many people have good number sense when they deal with money. Think about adding 99 cents and 1 cent. Do you see how this applies to adding 9.99 + 1.01?
Simplify: (5.58c + 8.75c) + 1.25c.
- Answer
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\(15.58c\)
Simplify: (8.79d + 3.55d) + 5.45d.
- Answer
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\(17.79d\)
No matter what you are doing, it is always a good idea to think ahead. When simplifying an expression, think about what your steps will be. The next example will show you how using the Associative Property of Multiplication can make your work easier if you plan ahead.
Simplify the expression: [1.67(8)](0.25).
Solution
Notice that multiplying (8)(0.25) is easier than multiplying 1.67(8) because it gives a whole number. (Think about having 8 quarters—that makes $2.)
| Regroup. | 1.67[(8)(0.25)] |
| Multiply in the brackets first. | 1.67[2] |
| Multiply. | 3.34 |
Simplify: [1.17(4)](2.25).
- Answer
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\(10.53\)
Simplify: [3.52(8)](2.5).
- Answer
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\(70.4\)
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.
Simplify: 6(9x).
Solution
| Use the associative property of multiplication to re-group. | (6 • 9)x |
| Multiply in the parentheses. | 54x |
Simplify: 8(3y).
- Answer
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\(24y\)
Simplify: 12(5z).
- Answer
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\(60z\)
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 3x + 7 + 4x + 5 by rewriting it as 3x + 4x + 7 + 5 and then simplified it to 7x + 12. We were using the Commutative Property of Addition.
Simplify: 18p + 6q + (−15p) + 5q.
Solution
Use the Commutative Property of Addition to re-order so that like terms are together.
| Re-order terms. | 18p + (−15p) + 6q + 5q |
| Combine like terms. | 3p + 11q |
Simplify: 23r + 14s + 9r + (−15s).
- Answer
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\(32r-s\)
Simplify: 37m + 21n + 4m + (−15n).
- Answer
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\(41m+6n\)
Practice Makes Perfect
Use the Commutative and Associative Properties
In the following exercises, use the commutative properties to rewrite the given expression.
- 8 + 9 = ___
- 7 + 6 = ___
- 8(−12) = ___
- 7(−13) = ___
- (−19)(−14) = ___
- (−12)(−18) = ___
- −11 + 8 = ___
- −15 + 7 = ___
- x + 4 = ___
- y + 1 = ___
- −2a = ___
- −3m = ___
In the following exercises, use the associative properties to rewrite the given expression.
- (11 + 9) + 14 = ___
- (21 + 14) + 9 = ___
- (12 · 5) • 7 = ___
- (14 · 6) • 9 = ___
- (−7 + 9) + 8 = ___
- (−2 + 6) + 7 = ___
- \(\left(16 \cdot \dfrac{4}{5}\right) • 15 = ___
- \(\left(13 \cdot \dfrac{2}{3}\right) • 18 = ___
- 3(4x) = ___
- 4(7x) = ___
- (12 + x) + 28 = ___
- (17 + y) + 33 = ___
Evaluate Expressions using the Commutative and Associative Properties
In the following exercises, evaluate each expression for the given value.
- If y = \(\dfrac{5}{8}\), evaluate:
- y + 0.49 + (− y)
- y + (− y) + 0.49
- If z = \(\dfrac{7}{8}\), evaluate:
- z + 0.97 + (− z)
- z + (− z) + 0.97
- If c = \(− \dfrac{11}{4}\), evaluate:
- c + 3.125 + (− c)
- c + (− c) + 3.125
- If d = \(− \dfrac{9}{4}\), evaluate:
- d + 2.375 + (− d)
- d + (− d) + 2.375
- If j = 11, evaluate:
- \(\dfrac{5}{6} \left(\dfrac{6}{5} j \right)\)
- \(\left(\dfrac{5}{6} \cdot \dfrac{6}{5}\right)j\)
- If k = 21, evaluate:
- \(\dfrac{4}{13} \left(\dfrac{13}{4}k \right)\)
- \(\left(\dfrac{4}{13} \cdot \dfrac{13}{4}\right)k\)
- If m = −25, evaluate:
- \(- \dfrac{3}{7} \left(\dfrac{7}{3}m \right)\)
- \(\left(- \dfrac{3}{7} \cdot \dfrac{7}{3}\right)m\)
- If n = −8, evaluate:
- \(- \dfrac{5}{21} \left(\dfrac{21}{5}n \right)\)
- \(\left(- \dfrac{5}{21} \cdot \dfrac{21}{5}\right)n\)
Simplify Expressions Using the Commutative and Associative Properties
In the following exercises, simplify.
- −45a + 15 + 45a
- 9y + 23 + (−9y)
- \(\dfrac{1}{2} + \dfrac{7}{8} + \left(− \dfrac{1}{2}\right)\)
- \(\dfrac{2}{5} + \dfrac{5}{12} + \left(− \dfrac{2}{5}\right)\)
- \(\dfrac{3}{20} \cdot \dfrac{49}{11} \cdot \dfrac{20}{3}\)
- \(\dfrac{13}{18} \cdot \dfrac{25}{7} \cdot \dfrac{18}{13}\)
- \(\dfrac{7}{12} \cdot \dfrac{9}{17} \cdot \dfrac{24}{7}\)
- \(\dfrac{3}{10} \cdot \dfrac{13}{23} \cdot \dfrac{50}{3}\)
- −24 • 7 • \(\dfrac{3}{8}\)
- −36 • 11 • \(\dfrac{4}{9}\)
- \(\left(\dfrac{5}{6} + \dfrac{8}{15}\right) + \dfrac{7}{15}\)
- \(\left(\dfrac{1}{12} + \dfrac{4}{9}\right) + \dfrac{5}{9}\)
- \(\dfrac{5}{13} + \dfrac{3}{4} + \dfrac{1}{4}\)
- \(\dfrac{8}{15} + \dfrac{5}{7} + \dfrac{2}{7}\)
- (4.33p + 1.09p) + 3.91p
- (5.89d + 2.75d) + 1.25d
- 17(0.25)(4)
- 36(0.2)(5)
- [2.48(12)](0.5)
- [9.731(4)](0.75)
- 7(4a)
- 9(8w)
- −15(5m)
- −23(2n)
- 12\(\left(\dfrac{5}{6} p\right)\)
- 20\(\left(\dfrac{3}{5} q\right)\)
- 14x + 19y + 25x + 3y
- 15u + 11v + 27u + 19v
- 43m + (−12n) + (−16m) + (−9n)
- −22p + 17q + (−35p) + (−27q)
- \(\dfrac{3}{8}g + \dfrac{1}{12}h + \dfrac{7}{8}g + \dfrac{5}{12}h\)
- \(\dfrac{5}{6}a + \dfrac{3}{10}b + \dfrac{1}{6}a + \dfrac{9}{10}b\)
- 6.8p + 9.14q + (−4.37p) + (−0.88q)
- 9.6m + 7.22n + (−2.19m) + (−0.65n)
Everyday Math
- Stamps Allie and Loren need to buy stamps. Allie needs four $0.49 stamps and nine $0.02 stamps. Loren needs eight $0.49 stamps and three $0.02 stamps.
- How much will Allie’s stamps cost?
- How much will Loren’s stamps cost?
- What is the total cost of the girls’ stamps?
- How many $0.49 stamps do the girls need altogether? How much will they cost?
- How many $0.02 stamps do the girls need altogether? How much will they cost?
- Counting Cash Grant is totaling up the cash from a fundraising dinner. In one envelope, he has twenty-three $5 bills, eighteen $10 bills, and thirty-four $20 bills. In another envelope, he has fourteen $5 bills, nine $10 bills, and twenty-seven $20 bills.
- How much money is in the first envelope?
- How much money is in the second envelope?
- What is the total value of all the cash?
- What is the value of all the $5 bills?
- What is the value of all $10 bills?
- What is the value of all $20 bills?
Writing Exercises
- In your own words, state the Commutative Property of Addition and explain why it is useful.
- In your own words, state the Associative Property of Multiplication and explain why it is useful.
Self Check
(a) After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
(b) After reviewing this checklist, what will you do to become confident for all objectives?
Contributors and Attributions
Lynn Marecek (Santa Ana College) and MaryAnne Anthony-Smith (Formerly of Santa Ana College). This content is licensed under Creative Commons Attribution License v4.0 "Download for free at http://cnx.org/contents/fd53eae1-fa2...49835c3c@5.191."