: If n is any positive
integer and a is any real number, a
= a · a
· a · a · · · · a: If n is any positive
integer and a is any real number, a= a · a
· a · a · · · · awhere the factor a occurs n times.
Definition of a
: For any nonzero real number
a, a
= 1.
Definition of a
: If a is a nonzero real number
and n is any integer, then a
= 
Definition of a
: If n is an even positive
integer, and if a>0, then a
is the positive
real
number whose nth power is a. That is, (a
)
= a.
If n is an odd positive integer, and if a is any real number,
then a is
the positive or negative real number whose nth power is a. That is,
(a
)
= a.
Definition of a
: For all integers m, all positive
integers n, and all real numbers a for
which a
is a real number, a
= (a
)
Example
Evaluate: a) 3
b) -2
c) (-5)
d) (7x)
e) -9
f) (-5)
g) 2
h) 81
i) (-32)
Solution:
a) 3
= 3·3·3·3 = 81
b) -2
= -2·2·2·2·2·2
= -64
c) (-5)
= (-5)(-5)(-5) = -125
d) (7x)
= 1
e) -9
= -1
f) (-5)
= 
= 
= 
g) 2
= 
= 
= 
h) 81
= (81
)
= 3
= 27
since 
i) (-32)
= 
= 
= 
= 
(-32)
= -2
since (-2)
= -32
Rules of Exponents
For all rational numbers r and s, and for all positive numbers a and b:
Example
Use rules of exponents to simplify each expression. Write answers without negative exponents. Assume that all variables represent nonzero real numbers.
a) 
b) 
c) 
d) 
e) 
f) 
Solution:
a) 
b) 
c) 
d) 
e) 
f) 