32 CHAPTER 1 Review of the Real Number System
OBJECTIVES The basic properties of real numbers studied in this section reflect results that occur
consistently in work with numbers. They have been generalized to apply to expres-
sions with variables as well.
OBJECTIVE 1 Use the distributive property.Notice that
and
so
This idea is illustrated by the divided rectangle in FIGURE 18. Similarly,
and
so
These examples are generalized to allreal numbers as the distributive property
of multiplication with respect to addition,or simply the distributive property.
- 435 + 1 - 324 = - 4152 + 1 - 421 - 32.
- 4152 + 1 - 421 - 32 =- 20 + 12 = - 8 ,
- 435 + 1 - 324 =- 4122 = - 8
213 + 52 = 2 # 3 + 2 # 5.
2 # 3 + 2 # 5 = 6 + 10 = 16 ,
213 + 52 = 2 # 8 = 16
Properties of Real Numbers
1.4
1 Use the distributive
property.
2 Use the identity
properties.
3 Use the inverse
properties.
4 Use the
commutative and
associative
properties.
5 Use the
multiplication
property of 0.Distributive PropertyFor any real numbers a, b, and c, the following are true.
a 1 bc 2 abac and 1 bc 2 abaca
Area of left part is 2. 3 = 6.
Area of right part is 2. 5 = 10.
3 52 2Area of total rectangle is 2(3 + 5) = 16.
FIGURE 18
The distributive property can also be written in “reverse” as
and.
It can be extended to more than two numbers as well.
The distributive property provides a way to rewrite a product as a sum
. It is also used to write a sum as a product.
NOTE When we rewrite as we sometimes refer to the process as
“removing” or “clearing” parentheses.
a 1 b+ c 2 ab+ac,
abac
a 1 bc 2
a 1 bcd 2 abacad
abaca 1 bc 2 baca 1 bc 2 a
Using the Distributive PropertyUse the distributive property to rewrite each expression.
(a)
= 3 x+ 3 y
31 x+ y 2
EXAMPLE 1
Use the first form of the property to
rewrite the given product as a sum.(b)
=- 10 - 2 k
= - 2152 + 1 - 221 k 2
- 215 +k 2 (c)
= 12 x
= 14 + 82 x
4 x+ 8 x Use the distributive property
in reverse to rewrite the
given sum as a product.