Intermediate Algebra (11th edition)

SECTION 1.4 Properties of Real Numbers 35

and

The way in which the numbers being added or multiplied are grouped does not matter.

The same answers result.

These arithmetic examples can be extended to algebra.

517 # 22 = 5 # 14 =70.

15 # 72 # 2 = 35 # 2 = 70

Commutative and Associative Properties

For any real numbers a, b, and c, the following are true.

Commutative properties

(The orderof the two terms or factors changes.)

Associative properties

(The groupingamong the three terms or factors changes, but the order stays the

same.)

a 1 bc 2  1 ab 2 c

a 1 bc 2  1 ab 2 c

abba

abba

⎧

⎨

⎩

⎧

⎨

⎩

The commutative properties are used to change the order of the terms or factors

in an expression. Think of commutingfrom home to work and then from work to home.

The associative properties are used to regroup the terms or factors of an expression. The

grouped terms or factors are associated.

Using the Commutative and Associative Properties

Simplify.

Order of operations

Distributive property

Add inside parentheses.

The next step would be to add 3mand 3, but they are unlike terms. To get 3mand

together, we use the associative and commutative properties, inserting paren-

theses and brackets according to the order of operations.

Associative property

Commutative property

Associative property

Combine like terms.

Associative property

Add.

In practice, many of these steps are not written down, but you should realize that

the commutative and associative properties are used whenever the terms in an ex-

pression are rearranged to combine like terms. NOW TRY

=- 3 m+ 11

=- 3 m+ 13 + 82

= 1 - 3 m+ 32 + 8

= 313 m+ 3 - 6 m 42 + 34 + 8

= 33 m+ 1 - 6 m+ 324 + 8

= 33 m+ 13 - 6 m 24 + 8

= 313 m+ 32 - 6 m 4 + 8

- 6 m

= 3 m+ 3 - 6 m+ 8

= 1 - 2 + 52 m+ 3 - 6 m+ 8

= 1 - 2 m+ 5 m 2 + 3 - 6 m+ 8

- 2 m+ 5 m+ 3 - 6 m+ 8

NOW TRY EXAMPLE 3

EXERCISE 3
Simplify.

• 7 x+ 10 - 3 x- 4 +x

NOW TRY ANSWER

3. 
   
   
   
   • 9 x+ 6