7.2 Commutative and associative properties  (Page 4/4)

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Simplify the expression: $\left[1.67\left(8\right)\right]\left(0.25\right).$

Solution

Notice that multiplying $\left(8\right)\left(0.25\right)$ is easier than multiplying $1.67\left(8\right)$ because it gives a whole number. (Think about having $8$ quarters—that makes $\text{2.)}$

 $\left[1.67\left(8\right)\right]\left(0.25\right)$ Regroup. $1.67\left[\left(8\right)\left(0.25\right)\right]$ Multiply in the brackets first. $1.67\left[2\right]$ Multiply. $3.34$

Simplify: $\left[1.17\left(4\right)\right]\left(2.25\right).$

10.53

Simplify: $\left[3.52\left(8\right)\right]\left(2.5\right).$

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: $6\left(9x\right).$

Solution

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

Simplify: $8\left(3y\right).$

24 y

Simplify: $12\left(5z\right).$

60 z

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: $18p+6q+\left(-15p\right)+5q.$

Solution

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

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

Simplify: $23r+14s+9r+\left(-15s\right).$

32 r s

Simplify: $37m+21n+4m+\left(-15n\right).$

41 m + 6 n

Key concepts

• Commutative Properties
• If $a,b$ are real numbers, then $a+b=b+a$
• Commutative Property of Multiplication:
• If $a,b$ are real numbers, then $a\cdot b=b\cdot a$
• Associative Properties
• If $a,b,c$ are real numbers then $\left(a+b\right)+c=a+\left(b+c\right)$
• Associative Property of Multiplication:
• If $a,b,c$ are real numbers then $\left(a\cdot b\right)\cdot c=a\cdot \left(b\cdot c\right)$

Practice makes perfect

Use the Commutative and Associative Properties

In the following exercises, use the commutative properties to rewrite the given expression.

$8+9=___$

$7+6=___$

7 + 6 = 6 + 7

$8\left(-12\right)=___$

$7\left(-13\right)=___$

7(−13) = (−13)7

$\left(-19\right)\left(-14\right)=___$

$\left(-12\right)\left(-18\right)=___$

(−12)(−18) = (−18)(−12)

$-11+8=___$

$-15+7=___$

−15 + 7 = 7 + (−15)

$x+4=___$

$y+1=___$

y + 1 = 1 + y

$-2a=___$

$-3m=___$

−3 m = m (−3)

In the following exercises, use the associative properties to rewrite the given expression.

$\left(11+9\right)+14=___$

$\left(21+14\right)+9=___$

(21 + 14) + 9 = 21 + (14 + 9)

$\left(12·5\right)·7=___$

$\left(14·6\right)·9=___$

(14 · 6) · 9 = 14(6 · 9)

$\left(-7+9\right)+8=___$

$\left(-2+6\right)+7=___$

(−2 + 6) + 7 = −2 + (6 + 7)

$\left(16·\frac{4}{5}\right)·15=___$

$\left(13·\frac{2}{3}\right)·18=___$

$\left(13·\frac{2}{3}\right)·18=13\left(\frac{2}{3}·18\right)$

$3\left(4x\right)=___$

$4\left(7x\right)=___$

4(7 x ) = (4 · 7) x

$\left(12+x\right)+28=___$

$\left(17+y\right)+33=___$

(17 + y ) + 33 = 17 + ( y + 33)

Evaluate Expressions using the Commutative and Associative Properties

In the following exercises, evaluate each expression for the given value.

If $y=\frac{5}{8},$ evaluate:

1. $\phantom{\rule{0.2em}{0ex}}y+0.49+\left(-y\right)$
2. $\phantom{\rule{0.2em}{0ex}}y+\left(-y\right)+0.49$

If $z=\frac{7}{8},$ evaluate:

1. $\phantom{\rule{0.2em}{0ex}}z+0.97+\left(-z\right)$
2. $\phantom{\rule{0.2em}{0ex}}z+\left(-z\right)+0.97$

1. 0.97
2. 0.97

If $c=-\frac{11}{4},$ evaluate:

1. $\phantom{\rule{0.2em}{0ex}}c+3.125+\left(-c\right)$
2. $\phantom{\rule{0.2em}{0ex}}c+\left(-c\right)+3.125$

If $d=-\frac{9}{4},$ evaluate:

1. $\phantom{\rule{0.2em}{0ex}}d+2.375+\left(-d\right)$
2. $\phantom{\rule{0.2em}{0ex}}d+\left(-d\right)+2.375$

1. 2.375
2. 2.375

If $j=11,$ evaluate:

1. $\phantom{\rule{0.2em}{0ex}}\frac{5}{6}\left(\frac{6}{5}j\right)$
2. $\phantom{\rule{0.2em}{0ex}}\left(\frac{5}{6}·\frac{6}{5}\right)j$

If $k=21,$ evaluate:

1. $\phantom{\rule{0.2em}{0ex}}\frac{4}{13}\left(\frac{13}{4}k\right)$
2. $\phantom{\rule{0.2em}{0ex}}\left(\frac{4}{13}·\frac{13}{4}\right)k$

1. 21
2. 21

If $m=-25,$ evaluate:

1. $\phantom{\rule{0.2em}{0ex}}-\frac{3}{7}\left(\frac{7}{3}m\right)$
2. $\phantom{\rule{0.2em}{0ex}}\left(-\frac{3}{7}·\frac{7}{3}\right)m$

If $n=-8,$ evaluate:

1. $\phantom{\rule{0.2em}{0ex}}-\frac{5}{21}\left(\frac{21}{5}n\right)$
2. $\phantom{\rule{0.2em}{0ex}}\left(-\frac{5}{21}·\frac{21}{5}\right)n$

1. −8
2. −8

Simplify Expressions Using the Commutative and Associative Properties

In the following exercises, simplify.

$-45a+15+45a$

$9y+23+\left(-9y\right)$

23

$\frac{1}{2}+\frac{7}{8}+\left(-\frac{1}{2}\right)$

$\frac{2}{5}+\frac{5}{12}+\left(-\frac{2}{5}\right)$

$\frac{5}{12}$

$\frac{3}{20}·\frac{49}{11}·\frac{20}{3}$

$\frac{13}{18}·\frac{25}{7}·\frac{18}{13}$

$\frac{25}{7}$

$\frac{7}{12}·\frac{9}{17}·\frac{24}{7}$

$\frac{3}{10}·\frac{13}{23}·\frac{50}{3}$

$\frac{65}{23}$

$-24·7·\frac{3}{8}$

$-36·11·\frac{4}{9}$

−176

$\left(\frac{5}{6}+\frac{8}{15}\right)+\frac{7}{15}$

$\left(\frac{1}{12}+\frac{4}{9}\right)+\frac{5}{9}$

$\frac{13}{12}$

$\frac{5}{13}+\frac{3}{4}+\frac{1}{4}$

$\frac{8}{15}+\frac{5}{7}+\frac{2}{7}$

$\frac{23}{15}$

$\left(4.33p+1.09p\right)+3.91p$

$\left(5.89d+2.75d\right)+1.25d$

9.89 d

$17\left(0.25\right)\left(4\right)$

$36\left(0.2\right)\left(5\right)$

36

$\left[2.48\left(12\right)\right]\left(0.5\right)$

$\left[9.731\left(4\right)\right]\left(0.75\right)$

29.193

$7\left(4a\right)$

$9\left(8w\right)$

72 w

$-15\left(5m\right)$

$-23\left(2n\right)$

−46 n

$12\left(\frac{5}{6}p\right)$

$20\left(\frac{3}{5}q\right)$

12 q

$14x+19y+25x+3y$

$15u+11v+27u+19v$

42 u + 30 v

$43m+\left(-12n\right)+\text{​}\left(-16m\right)+\left(-9n\right)$

$-22p+17q+\left(-35p\right)+\left(-27q\right)$

−57 p + (−10 q )

$\frac{3}{8}g+\frac{1}{12}h+\frac{7}{8}g+\frac{5}{12}h$

$\frac{5}{6}a+\frac{3}{10}b+\frac{1}{6}a+\frac{9}{10}b$

$a+\frac{6}{5}b$

$6.8p+9.14q+\left(-4.37p\right)+\left(-0.88q\right)$

$9.6m+7.22n+\left(-2.19m\right)+\left(-0.65n\right)$

7.41 m + 6.57 n

Everyday math

Stamps Allie and Loren need to buy stamps. Allie needs four $\text{0.49}$ stamps and nine $\text{0.02}$ stamps. Loren needs eight $\text{0.49}$ stamps and three $\text{0.02}$ 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 $\text{0.49}$ stamps do the girls need altogether? How much will they cost?

How many $\text{0.02}$ 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 $\text{5}$ bills, eighteen $\text{10}$ bills, and thirty-four $\text{20}$ bills. In another envelope, he has fourteen $\text{5}$ bills, nine $\text{10}$ bills, and twenty-seven $\text{20}$ 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 $\text{5}$ bills?

What is the value of all $\text{10}$ bills?

What is the value of all $\text{20}$ bills?

1. $975 2.$700
3. $1675 4.$185
5. $270 6.$1220

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

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

After reviewing this checklist, what will you do to become confident for all objectives?

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