Rules for Integral Exponents
Negative exponents are used to make expressions involving reciprocals
simpler looking and easier to write. Negative exponents have the added benefit
of working in conjunction with all of the rules of exponents that you should
have learned by now. For example, we can use the product rule to get
x -2 · x -3 = x -2 + (-3) = x -5
and the quotient rule to get
With negative exponents there is no need to state the quotient rule in two
parts. It can be stated simply as
for any integers m and n. We list the rules of exponents here for easy
reference.
Rules for Integral Exponents
The following rules hold for nonzero real numbers a and b and any integers m
and n.
| 1) a0 = 1 |
Definition of zero exponent |
| 2) am · an = am + n |
Product rule |
| 3)
|
Quotient rule |
| 4) (am )n = amn |
Power rule |
| 5) (ab)n = an bn |
Power of a product rule |
6)
 |
Power of a quotient rule |
The definitions of the different types of exponents are a really clever
mathematical invention. the fact that we have rules for performing arithmetic
with those exponents makes the notation of exponents even more amazing.
Example 1
The product and quotient rules for integral exponents
Simpify. Write your answers without negative exponents. Assume that the
variables represent nonzero real numbers.
Solution
| a) b-3b5 |
= b-3 + 5 |
Product rule |
| |
= b2 |
Simplify |
b)
 |
Product rule Definition of negative exponent |
c)
 |
Quotient rule Simplify.
Definition of negative exponent |
c)
 |
|