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Integer Exponents and the Quotient Rule

1. Zero Exponent: For any nonzero real number a, a⁰ =1.

2. Negative Exponents: For any nonzero real number a and any
integer n,.

Note that the negative exponent does not change the sign of the base.
(It is not a coefficient.)

3. Note that this means: , for a ≠ 0.

4. Changing from Negative to Positive Exponents: For any nonzero
numbers a and b, and any integers m and n,

Note that these rules do not work if there is a sum or difference in the
numerator or the denominator.

5. Quotient Rule for Exponents: For any nonzero real number a and
any integers m and n,

When dividing like bases, keep the same base and subtract the
exponents. (You can always use the definition of an exponent and
cancel common factors.)

6. These rules are often combined. Also note that these rules can be
combined in several correct orders.

7. See the summary of rules on pages 268 and 269.