Section 5.2Zero and Negative

Exponents

The Zero Exponent Rule

Zero Exponents: Any nonzero base raised to the 0 power is 1.

For any nonzero real number x, x0 = 1.

Why?

Ex: (a) x0 = 1

(b) 2x0 = 2 ∙ x0 = 2 ∙ 1 = 2

(c) (a2 b)0 = 1

19

9

3

32

2

1333

3 022

2

2

The Negative Integer Exponent Rule

Negative Exponents: For any nonzero real number x and any integer n,

In words, x−n is the reciprocal of xn.

Ex: Express using positive exponents and simplify:

7

6

7

7

7

11

7

177)(

21222)(

64

1

4

14)(

01

44

44

3

c

xxxxb

a

n

n

xx

1

16

1

)2)(2)(2)(2(

1

)2(

1)2()(

16

1

16

11

2

11212)(

4

4

4

44

e

d

Change Negative Exponents in Fractions to

Positive Exponents

We can obtain this result in a simpler way. If there is a negative

exponent, move it to the other side of the fraction and make the

exponent positive.

m

n

n

mn

n x

y

y

xandx

x

1

For any nonzero real numbers x and y, and any integers m and n

How does this work?

Ex:

7

2

2

7

4

4

66)(

1)(

x

y

y

xb

bb

a

Change Negative Exponents in Fractions to Positive

Exponents

For any nonzero real numbers x and y, and any integer n,

Why does this work?

Ex:

nn

x

y

y

x

44

m

n

n

m

Ex: Simplify.

All Exponent Rules to Simplify Expressions

The rules for exponents involving products,

powers, and quotients are also true for zero and

negative exponents.

The rules for exponents are used to simplify

expressions involving products, quotients,

and powers.

In general, an expression involving

exponents is simplified when:

Each base occurs only once.

No powers are raised to powers

There are no parentheses.

There are no negative or zero exponents.

Ex: Simplify. Do not use negative

exponents.

62323

22

3

23

30

213021)3(103731037

310731034373)3(4

3

73

34

2

2224224

421

242

422

241

44

)(

)2(2

2)(

)(

2

9

2

9

2

9

2

3

3

2)(

pppp

pc

s

rsrsrsr

srsrsrsr

srb

b

ababa

ba

ba

ba

baa