Negative & Zero Exponents

Why a^0 = 1 and a^-n = 1/a^n, and rewriting with positive exponents.

The idea

Zero and negative exponents extend the meaning of a power beyond the cases the definition covers. Defining $a^{n}$ as $n$ copies of $a$ multiplied together makes sense only when $n$ is a positive whole number, so $a^{0}$ and $a^{-3}$ as yet mean nothing. We are free to define them, and the principled choice is whatever keeps the rules already established working.

The rule that decides is the quotient rule, $\frac{a^{m}}{a^{n}} = a^{m-n}.$ Applied with $m = n$, the rule gives $a^{0}$, while the fraction itself, a nonzero number divided by itself, equals $1$. Applied with $m$ smaller than $n$, the rule gives a negative exponent, while cancelling by hand leaves the surplus factors of $a$ in the denominator. Each comparison forces exactly one meaning: $a^{0} = 1, \qquad a^{-n} = \frac{1}{a^{n}} \qquad (a \neq 0).$ These are the only definitions under which the earlier rules continue to hold. One caution: a negative exponent produces a reciprocal, never a negative value.

Ways to work on it

• Walkthrough. Derive the zero and negative-exponent definitions from the quotient rule.
• Practice. Evaluate a zero or negative-exponent power to a single number.
• Hardest. Simplify an expression to use only positive exponents.

Not sure where to start? Take the ten-question placement test.