64 CHAPTER 1 The Real Number System
The distributive propertysays that multiplying a number aby a sum of num-
bers gives the same result as multiplying aby band aby cand then adding the
two products.
b+ c
Distributive Propertya 1 bc 2 abac and 1 bc 2 abaca
NOW TRY
EXERCISE 8
Simplify.
1
3
x+ 7 +1
3
xNOW TRY ANSWER
- 7
Using Properties to Simplify an ExpressionSimplify.
Order of operations
Commutative property
Associative property
Inverse property= 10 Identity property NOW TRY
= 10 + 0
= 10 + 31 - 2 x 2 + 2 x 4
= 310 + 1 - 2 x 24 + 2 x
= 1 - 2 x+ 102 + 2 x
- 2 x+ 10 + 2 x
EXAMPLE 8
For anyvalue of
and 2xare
additive inverses.x, - 2 xNOTEThe steps of Example 8may be skipped when we actually do the simplification.
OBJECTIVE 5 Use the distributive property.The word distributemeans “to
give out from one to several.” Look at the value of the following expressions:
, which equals or 26
, which equals or 26.
Since both expressions equal 26,
This result is an example of the distributive property of multiplication with respect to ad-
dition,the only property involving bothaddition and multiplication. With this property,
a product can be changed to a sum or difference. This idea is illustrated in FIGURE 18.
215 + 82 = 2152 + 2182.
2152 + 2182 10 + 16,
215 + 82 21132 ,
The area of the left part is 2(5) = 10.
The area of the right part is 2(8) = 16.
The total area is 2(5 + 8) = 2(13) = 26,
or the total area is 2(5) + 2(8) = 10 + 16 = 26.
Thus, 2(5 + 8) = 2(5) + 2(8).25 82FIGURE 18As the arrows show, the aoutside the parentheses is “distributed” over the band cin-
side. The distributive property is also valid for multiplication over subtraction.
and
The distributive property can be extended to more than two numbers.
The distributive property can also be used “in reverse.”
acbc 1 ab 2 c
a 1 bcd 2 abacad
a 1 bc 2 abac 1 bc 2 abaca
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