Negative Integral Exponents
If x is nonzero, the reciprocal of x is written as
For example, the reciprocal of 2^{3} is written
To write the reciprocal of an exponential expression in a simpler way, we use a
negative exponent. So
In general we have the following definition.
Negative Integral Exponents
If a is a nonzero real number and n is a positive integer, then
(If n is positive, n is negative.)
Example 1
Simplifying expressions with negative exponents
Simplify.
a) 2^{5}
b) (2)^{5}
c)
Solution
a)
b)
Definition of negative exponent
c)
Caution
In simplifying 5^{2}, the negative sign preceding the 5 is used
after 5 is squared and the reciprocal is found. So
To evaluate a^{ n}, you can first find the nth power of a and then
the reciprocal. However, the result is the same if you first find the reciprocal
of a and then find the nth power of the reciprocal. For example,
So the power and the reciprocal can be found in either order. If the exponent
is 1, we simply find the reciprocal. For example,
Because 3^{ 2} Â· 3^{2} = 1, the reciprocal of 3^{ 2}
is 3^{2}, and we have
These expamples illustrate the following rules.
Rules for Negative Exponents
If a is a nonzero real number and n is a positive integer, then
Example 2
Using the rules for negative exponents
Simplify
Solution
