Zero and Negative Exponents
December 2010
Zero and Negative Exponents
Objective: The student will learn to simplify
expressions involving the exponent zero and
negative exponents.
Zero Exponents
The definition of power can be extended to
zero.
The First Law states,
aman = am+n
Suppose
n=0 
 a0 = 1
ama0 = am+0 = am
Zero Exponents
Any real number, except zero, raised to the
zero power equals one.
a  0 , a  1
0
Examples:
50 = 1
(6,372,961)0 = 1
(7)0 = 1
x  0  x0 = 1
Negative Exponents
The definition of power can be extended to
negative integers.
The First Law states,
aman = am+n
Consider
anan = an +n = a0 = 1

an

an =
is the reciprocal of an
1
an
Negative Exponents
Any real number, except zero, raised to a power of
negative n equals the reciprocal of that same number
raised to the power of n.
a  0 , a
Examples:
51 = 1/5
23 = 1/23 = 1/8
(3)3 = 1/27
x  0  x5 = 1/(x5)
n
1
 n
a
Laws of Exponents
All the Laws of Exponents hold even if some of
the exponents are negative or zero.
Example:
ca h
3 2
(Power of a Power)
6
a
f
a
a
3 2
Laws of Exponents
The 4th and 5th Laws of Exponents are equivalent:
Law 4:
Law 5:
2
a
1
27
5

a

a

7
5
a
a
2
a
1
1


7
72
5
a
a
a
Write in simplest form without
negative or zero exponents.
c3 5 h
1 2
b g c5 h
 3
2
1 2
1
2

2 5
3
b g
25

9
Write in simplest form without
negative or zero exponents.
0
3
3 x y
2 x 1 y 2

1 x
3( 1)
2
2
1x y

2
3
y
 2
2x
3
y
1( 2 )
Write in simplest form without
negative or zero exponents.
FG 2 x IJ
H 5y K
2
3
1
2x h
c

c5y h
2 1
3 1
1
2
2 x
 1 3
5 y
2
5x y

2
3
Write without using fractions.
3
10,000
3
 4
10
 3  104
Scientific Notation
Write without using fractions.
2
5x
3
yz
c h
 5x yz
2
3 1
 5x 2 y 1z 3
Class work: Oral Exercises p. 218: 1-16
Homework: p. 218: 1-51 odd, p 214: 33, 34