Beginning Algebra, 11th Edition

SECTION 1.7 Properties of Real Numbers^61

NOW TRY ANSWERS

(a) 7 (b)

(a)
(b) 35 # 1 - 424 # 9

1 - 9 + 32 + 7

5

OBJECTIVE 2 Use the associative properties. When we associateone object

with another, we think of those objects as being grouped together.

The associative propertiessay that when we add or multiply three numbers, we

can group the first two together or the last two together and get the same answer.

Associative Properties

Addition

1 ab 2 ca 1 bc 2 Multiplication

1 ab 2 ca 1 bc 2

Using the Associative Properties

Use an associative property to complete each statement.

(a)

Associative property of addition

(b)

Associative property of multiplication NOW TRY

32 # 1 - 724 # 6 = 2 # 31 - 72 # 64

32 # 1 - 724 # 6 = 2 #?

- 8 + 11 + 42 = 1 - 8 + 12 + 4

- 8 + 11 + 42 = 1 - 8 +? 2 + 4

NOW TRY EXAMPLE 2

EXERCISE 2
Use an associative property to
complete each statement.

(a)

(b) 531 - 42 # 94 =

- 9 + 13 + 72 =

The “order” is the same. The “grouping” changed.

Using the Commutative Properties

Use a commutative property to complete each statement.

(a)

Commutative property of addition

(b)

Commutative property of multiplication NOW TRY

- 2172 = 71 - 22

1 - 227 =? 1 - 22

- 8 + 5 = 5 + 1 - 82

- 8 + 5 = 5 +?

NOW TRY EXAMPLE 1

EXERCISE 1
Use a commutative property
to complete each statement.

(a)

(b) 1 - 524 = 4 #

7 + 1 - 32 =- 3 +

By the associative property, the sum (or product) of three numbers will be the

same no matter how the numbers are “associated” in groups. Parentheses can be left

out if a problem contains only addition (or multiplication). For example,

and can be written as

Distinguishing Between Properties

Is each statement an example of the associative or the commutative property?

(a)

The order of the three numbers is the same on both sides of the equals symbol.

The only change is in the grouping,or association, of the numbers. This is an exam-

ple of the associative property.

(b)

The same numbers, 3 and 10, are grouped on each side. On the left, the 3 appears

first, but on the right, the 10 appears first. Since the only change involves the order

of the numbers, this is an example of the commutative property.

6 # 13 # 102 = 6 # 110 # 32

12 + 42 + 5 = 2 + 14 + 52

EXAMPLE 3

1 - 1 + 22 + 3 - 1 + 12 + 32 - 1 + 2 + 3.

Notice that the “order” changed.