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학습 가이드 > Prealgebra

Evaluating and Simplifying Expressions Using the Commutative and Associative Properties

Learning Outcomes

  • Evaluate algebraic expressions for a given value using the commutative and associative properties of addition and multiplication
  • Simplify algebraic expressions using the commutative and associative properties of addition and multiplication
 

Evaluate Expressions using the Commutative and Associative Properties

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.

example

Evaluate each expression when x=78x=\frac{7}{8}.
  1. x+0.37+(−x)x+0.37+\left(-x\right)
  2. x+(−x)+0.37x+\left(-x\right)+0.37
Solution:
1.
x+0.37+(−−x)x+0.37+(--x)
Substitute 78\frac{7}{8} for xx . 78+0.37+(−−78)\color{red}{\frac{7}{8}}+0.37+(--\color{red}{\frac{7}{8}})
Convert fractions to decimals. 0.875+0.37+(−−0.875)0.875+0.37+(--0.875)
Add left to right. 1.245−−0.8751.245--0.875
Subtract. 0.370.37
2.
x+(−−x)+0.37x+(--x)+0.37
Substitute 78\frac{7}{8} for x. 78+(−−78)+0.37\color{red}{\frac{7}{8}}+(--\color{red}{\frac{7}{8}})+0.37
Add opposites first. 0.370.37
What was the difference between part 1 and part 2? Only the order changed. By the Commutative Property of Addition, x+0.37+(−x)=x+(−x)+0.37x+0.37+\left(-x\right)=x+\left(-x\right)+0.37. But wasn’t part 2 much easier?
 

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[ohm_question]145792[/ohm_question]
  Let’s do one more, this time with multiplication.

example

Evaluate each expression when n=17n=17. 1. 43(34n)\frac{4}{3}\left(\frac{3}{4}n\right) 2. (43⋅34)n\left(\frac{4}{3}\cdot \frac{3}{4}\right)n

Answer: Solution:

1.
43(34n)\frac{4}{3}(\frac{3}{4}n)
Substitute 17 for n. 43(34⋅17)\frac{4}{3}(\frac{3}{4}\cdot\color{red}{17})
Multiply in the parentheses first. 43(514)\frac{4}{3}(\frac{51}{4})
Multiply again. 1717
2.
(43⋅34)n(\frac{4}{3}\cdot\frac{3}{4})n
Substitute 17 for n. (43⋅34)⋅17(\frac{4}{3}\cdot\frac{3}{4})\cdot\color{red}{17}
Multiply. The product of reciprocals is 1. (1)⋅17(1)\cdot17
Multiply again. 1717
What was the difference between part 1 and part 2 here? Only the grouping changed. By the Associative Property of Multiplication, 43(34n)=(43⋅34)n\frac{4}{3}\left(\frac{3}{4}n\right)=\left(\frac{4}{3}\cdot \frac{3}{4}\right)n. By carefully choosing how to group the factors, we can make the work easier.

 

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[ohm_question]145796[/ohm_question]
 

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 the first example, part 2 was easier to simplify than part 1 because the opposites were next to each other and their sum is 00. Likewise, part 2 in the second example was easier, with the reciprocals grouped together, because their product is 11. In the next few examples, we’ll use our number sense to look for ways to apply these properties to make our work easier.  

example

Simplify: −84n+(−73n)+84n-84n+\left(-73n\right)+84n

Answer: Solution: Notice the first and third terms are opposites, so we can use the commutative property of addition to reorder the terms.

−84n+(−73n)+84n-84n+\left(-73n\right)+84n
Re-order the terms. −84n+84n+(−73n)-84n+84n+\left(-73n\right)
Add left to right. 0+(−73n)0+\left(-73n\right)
Add. −73n-73n

 

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[ohm_question]145797[/ohm_question]
Watch the following video for more similar examples of how to use the associative and commutative properties to simplify expressions. https://youtu.be/8vMRywgaqOE Now we will see how recognizing reciprocals is helpful. Before multiplying left to right, look for reciprocals—their product is 11.

example

Simplify: 715⋅823⋅157\frac{7}{15}\cdot \frac{8}{23}\cdot \frac{15}{7}

Answer: Solution: Notice the first and third terms are reciprocals, so we can use the Commutative Property of Multiplication to reorder the factors.

715⋅823⋅157\frac{7}{15}\cdot \frac{8}{23}\cdot \frac{15}{7}
Re-order the terms. 715⋅157⋅823\frac{7}{15}\cdot \frac{15}{7}\cdot \frac{8}{23}
Multiply left to right. 1⋅8231\cdot \frac{8}{23}
Multiply. 823\frac{8}{23}

 

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[ohm_question]145798[/ohm_question]
  In expressions where we need to add or subtract three or more fractions, combine those with a common denominator first.  

example

Simplify: (513+34)+14\left(\frac{5}{13}+\frac{3}{4}\right)+\frac{1}{4}

Answer: Solution: Notice that the second and third terms have a common denominator, so this work will be easier if we change the grouping.

(513+34)+14\left(\frac{5}{13}+\frac{3}{4}\right)+\frac{1}{4}
Group the terms with a common denominator. 513+(34+14)\frac{5}{13}+\left(\frac{3}{4}+\frac{1}{4}\right)
Add in the parentheses first. 513+(44)\frac{5}{13}+\left(\frac{4}{4}\right)
Simplify the fraction. 513+1\frac{5}{13}+1
Add. 15131\frac{5}{13}
Convert to an improper fraction. 1813\frac{18}{13}

 

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[ohm_question]145799[/ohm_question]
  When adding and subtracting three or more terms involving decimals, look for terms that combine to give whole numbers.

example

Simplify: (6.47q+9.99q)+1.01q\left(6.47q+9.99q\right)+1.01q

Answer: Solution: Notice that the sum of the second and third coefficients is a whole number.

(6.47q+9.99q)+1.01q\left(6.47q+9.99q\right)+1.01q
Change the grouping. 6.47q+(9.99q+1.01q)6.47q+\left(9.99q+1.01q\right)
Add in the parentheses first. 6.47q+(11.00q)6.47q+\left(11.00q\right)
Add. 17.47q17.47q

Many people have good number sense when they deal with money. Think about adding 9999 cents and 11 cent. Do you see how this applies to adding 9.99+1.01?9.99+1.01?

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[ohm_question]145800[/ohm_question]
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.

example

Simplify: 6(9x)6\left(9x\right)

Answer: Solution:

6(9x)6\left(9x\right)
Use the associative property of multiplication to re-group. (6⋅9)x\left(6\cdot 9\right)x
Multiply in the parentheses. 54x54x

 

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[ohm_question]145973[/ohm_question]
  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+53x+7+4x+5 by rewriting it as 3x+4x+7+53x+4x+7+5 and then simplified it to 7x+127x+12. We were using the Commutative Property of Addition.  

example

Simplify: 18p+6q+(−15p)+5q18p+6q+\left(-15p\right)+5q

Answer: Solution: Use the Commutative Property of Addition to re-order so that like terms are together.

18p+6q+(−15p)+5q18p+6q+\left(-15p\right)+5q
Re-order terms. 18p+(−15p)+6q+5q18p+\left(-15p\right)+6q+5q
Combine like terms. 3p+11q3p+11q

 

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Simplify: 23r+14s+9r+(−15s)23r+14s+9r+\left(-15s\right) 32r−s32r−s Simplify: 37m+21n+4m+(−15n)37m+21n+4m+\left(-15n\right) 41m+6n41m+6n #146471
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