Basic Mathematics for College Students

648 Chapter 8 An Introduction to Algebra

SECTION 8.2

Simplifying Algebraic Expressions

Objectives

1 Simplify products.

2 Use the distributive property.

3 Identify like terms.

4 Combine like terms.

In algebra, we frequently replace one algebraic expression with another that is

equivalent and simpler in form. That process, called simplifying an algebraic

expression,often involves the use of one or more properties of real numbers.

Simplify products.

The commutative and associative properties of multiplication can be used to simplify

certain products. For example, let’s simplify.

Rewrite as.

Use the associative property of multiplication to group 4 with 8.

Do the multiplication within the parentheses.

We have found that. We say that and are equivalent expressions

because for each value of , they represent the same number. For example, ifx10,

both expressions have a value of 320. Ifx3, both expressions have a value of96.

If If

same result same result

Success Tip By the commutative property of multiplication, we can change

the orderof factors. By the associative property of multiplication, we can

change the groupingof factors.

 (^320)   (^96) 

8(40)  320 8(12)  96

8(4x)8[4( 10 )] 32 x32( 10 ) 8(4x)8[4( 3 )] 32 x32( 3 )

x 10 x 3

x

8(4x) 32 x 8(4x) 32 x

 32 x

(84)x

8(4x) 8 (4x) 4 x 4 x

8(4x)

1

Self Check 1

Simplify:

a.

b.

c.

d.

Now TryProblems 15, 25, 29, and 31

36 a

2

9

yb

2

3



3

2

m

4(6u)(2)

9  6 s

EXAMPLE (^1) Simplify:
a.9(3b) b. c. d. e.
StrategyWe will use the commutative and associative properties of multiplication
to reorder and regroup the factors in each expression.
WHYWe want to group all of the numerical factors of an expression together so
that we can find their product.
Solution
a.9(3b)  (9 3)b
 27 b Do the multiplication within the parentheses.
b.
Do the multiplication:.
c.
Multiply within the brackets.
 105 p Complete the multiplication within the brackets.
[21(5)]p
Use the commutative and associative
properties of multiplication to reorder
and regroup the factors.
3(7p)(5)[3(7)(5)]p
 90 a 15(6) 90
Use the commutative property of multiplication to
15 a(6)15(6)a (^) reorder the factors.
Use the associative property of multiplication to
regroup the factors.
35 a

4

5

xb

8

3



3

8

15 a(6) 3(7p)(5) r

1

3

5

 6

90

 

21

 5

105