2. Zero and Negative Exponents
Zero Exponent: any nonzero expression to the zero
power is 1.
a = 1, a ≠ 0
0
5 =1
0
Negative Exponent: any term to a negative power is
the reciprocal of that term with a positive power
−n 1 −3 1
a = n ,a ≠ 0 2 = 3
a 2
1 1
= a ,a ≠ 0
n
−4
=2 4
a −n 2

3. Example 1 Use definition of zero and negative exponents
1 1 Definition of negative exponents
a. 3 –2
= 2 =
3 9
b. ( – 7) 0 = 1 Definition of zero exponent
c. 1 –2 1 Definition of negative exponents
=
5 1 2
5
1 Evaluate power.
=
1
25

4. Example 1 Use definition of zero and negative exponents
Simplify by multiplying numerator
= 25
and denominator by 25.
1
d. 0
–5
= 5 (Undefined) a–n is defined only for a nonzero
0 number a.

5. Properties of Exponents
a m ×a n = a m+n Product of powers
(a )
m n
= a m×n Power of a power
( ab) = a m ×b m Power of a product
m
am
= a m−n , a ≠ 0
an Quotient of powers
m
a am
 ÷ = m ,b ≠ 0 Power of a quotient
b b
a 0 = 1, a ≠ 0 Zero exponent
1 1
a −n = n
,a ≠ 0 Negative Exponent = an , a ≠ 0
a a −n

6. Example 2 Evaluate exponential expressions
a. 6–4 • 64 = 6–4 + 4 Product of powers property
= 60 Add exponents.
=1 Definition of zero exponent
b. (4–2 ) 2
= 4–2 •
2
Power of a power property
= 4–4 Multiply exponents.
1 Definition of negative exponents
= 4
4
1 Evaluate power.
=
256

7. Example 2 Evaluate exponential expressions
c. 1
= 34 Definition of negative exponents
3–4
= 81 Evaluate power.
d. 5–1
= 5–1 – 2 Quotient of powers property
52
= 5–3 Subtract exponents.
1
= 3 Definition of negative exponents
5
1
= Evaluate power.
125

8. Example 3 Use properties of exponents
Simplify the expression. Write your answer using only
positive exponents.
a. ( 2xy–5 )3 = 23 • x3 • ( y–5 )3 Power of a product property
= 8 • x3 • y–15 Power of a power property
8x3 Definition of negative
= 15 exponents
y
( 2x )–2 y5 y5 Definition of negative
b. =
– 4x y
2 2 ( 2x )2(– 4x2y2 ) exponents

9. Example 3 Use properties of exponents
y5 Power of a product property
=
( 4x2 ) (– 4x2y2 )
y5 Regroup factors in
=
[4(– 4 )]( x2 • x2 )y2 denominator.
y5 Simplify denominator.
=
– 16x4y2
y3 Quotient of powers
= –
16x4 property

10. 8.3 Warm-Up
Evaluate the expression.
0
 −9
1. 7 −3 2. −6
7 ⋅7 4
3. 


16 
Simplify the expression. Write your answer using only
positive exponents.
4. ( 4g ) −3
5. r −2
s −4