Using the Quotient Rule

Simplify by writing with positive exponents. Assume that all variables represent

nonzero real numbers.

(a) (b)

(c) (d)

q^5

q-^3

=q^5 -^1 -^32 =q^8

5 -^3

5 -^7

= 5 -^3 -^1 -^72 = 54 = 625

42

49

= 42 -^9 = 4 -^7 =

1

47

58

56

= 58 -^6 = 52 = 25

EXAMPLE 4

SECTION 5.2 Integer Exponents and the Quotient Rule 307

Definitions and Rules for Exponents

For any integers mand n, the following are true. Examples

Product rule

Zero exponent

Negative

Quotient rule

Power rule (a)

Power rule (b)

Power rule (c)

Negative-to-

a

4

7

b

• 2

= a

7

4

b

2

a

a

b

b

m

a

b

a

b

m

2 -^4

5 -^3

=

53

24

1 a0, b 02

am

bn



bn

am

a

2

3

b

2

=

22

32

a 1 b 02

a

b

b

m



am

bm

1 ab 2 mambm 13 k 24 = 34 k^4

14223 = 42

# 3

1 am 2 namn = 46

22

25

= 22 -^5 = 2 -^3 =

1

23

1 a 02

am

an

amn

5 -^3 =

1

53

an 1 a 02

1

an

a^0  1 1 a 02 1 - 320 = 1

am#anamn 74 # 75 = 74 +^5 = 79

positive rules

Keep the same base.

Be careful with signs.

(e)

Quotient rule

Subtract.

or

x^2

9

=

x^2

32

,

= 3 -^2 x^2

= 32 -^4 #x^5 -^3

=

32

34

x

5

x^3

32 x^5

34 x^3

(f )

= 1 m+ n 22 , mZ-n

= 1 m+ n 2 -^2 +^4

= 1 m+ n 2 -^2 -^1 -^42

1 m+n 2 -^2

1 m+n 2 -^4

(g)

Negative-to-positive rule

Product rule NOW TRY

The definitions and rules for exponents are summarized here.

=

14 y^7

x^5

=

7 # 21 y^2 y^5

x^2 x^3

7 x-^3 y^2

2 -^1 x^2 y-^5

NOW TRY
EXERCISE 4
Simplify by writing with
positive exponents. Assume
that all variables represent
nonzero real numbers.

(a) (b)

(c)

(d)

52 xy-^3

3 -^1 x-^2 y^2

1 pZ-q 2

1 p+q 2 -^3

1 p+q 2 -^7

t^4

t-^5

63

64

NOW TRY ANSWERS

4. (a) (b)

(c) (d)

75 x^3

y^5

1 p+q 24

t^9

1

6

exponent