OBJECTIVE 5 Simplify exponential expressions.

270 CHAPTER 5 Exponents, Polynomials, and Polynomial Functions

Using the Definitions and Rules for Exponents

Simplify each exponential expression so that no negative exponents appear in the

final result. Assume that all variables represent nonzero real numbers.

(a)

Product rule

Add exponents.

or a-n=a^1 n

1

27

=

1

33

,

= 3 -^3

= 32 +^1 -^52

32 # 3 -^5

EXAMPLE 8

(b)

Product rule

Add exponents.

= a-n=a^1 n

1

x^5

= x-^5

= x-^3 +^1 -^42 +^2

x-^3 #x-^4 #x^2

(c)

Power rule (a)

= 410 Multiply exponents.

= 41 -^221 -^52

14 -^22 -^5 (d)

Power rule (a)

Multiply exponents.

= a-n=a^1 n

1

x^24

=x-^24

=x^1 -^426

1 x-^426

(e)

Quotient rule

= a-n=a^1 n

y^7

x^6

= x-^6 y^7

= x-^4 -^2 #y^2 -^1 -^52

=

x-^4

x^2

y

2

y-^5

x-^4 y^2

x^2 y-^5

(f )

Power

rule (b)

or a-n=a^1 n

x^4

64

=

x^4

26

,

= 2 -^6 x^4

= 1232 -^2 # 1 x-^22 -^2

123 x-^22 -^2

(g)

Power rule (a)

Quotient rule

= a-n=a^1 n

9 x

4 y^4

=

9

4

x^1 y-^4

=

9

4

x^4 -^3 y-^2 -^2

=

9 x^4

y^2

#y

• 2

4 x^3

=

321 x^222

y^2

#y

• 2

4 x^3

a

3 x^2

y

b

2

a

4 x^3

y-^2

b

• 1

Combination

of rules

(h)

= 1 - 622 = 36

36

m^8 n^22

bm

=an

a-n

= b-m

1 - 622

m^8 n^22

=

m-^8 n-^22

1 - 62 -^2

=

1 m^42 -^21 n^112 -^2

1 - 62 -^2

= a

m^4 n^11

- 6

b

• 2

= a

m^5 -^1 n^4 - (-7)

- 6

b

• 2

a

- 4 m^5 n^4

24 mn-^7

b

• 2

Quotient

rule; divide

coefficients.

Subtract

exponents.

Power

rules (b)

and (c)

Power

rule (a)

The sign on

does notchange

in this step.

• 6

NOW TRY

NOW TRY
EXERCISE 8
Simplify. Assume that all
variables represent nonzero
real numbers.

(a)

(b)

(c)

(d) a

2 x^2

y^2

b

3

a

• 5 x-^2
  y

b

• 2

p^2 q-^4

p-^2 q-^1

15 -^322

x-^8 #x#x^4

NOW TRY ANSWERS

8. (a) (b)

(c) (d)

8 x^10

25 y^4

p^4

q^3

1

56

1

x^3

Subtract

exponents.