Definition of Powers with Negative Exponents
🎯 Big Idea
A negative exponent doesn't make the number negative — it tells us to take the reciprocal of the base raised to the positive version of that exponent.
📐 Formal Definition
For any nonzero number $a$ and any positive integer $n$:
$$a^{-n} = \frac{1}{a^n}$$
In words: $a$ raised to the power of $-n$ equals one divided by $a$ raised to the power of $n$.
⚠️ Important Restriction: The base $a$ cannot be zero. Why? Because $0^{-n} = \frac{1}{0^n} = \frac{1}{0}$, which is undefined (division by zero!).
Quick Reference
| Expression | Value |
|---|---|
| $2^{-1}$ | $\frac{1}{2}$ |
| $2^{-2}$ | $\frac{1}{4}$ |
| $2^{-3}$ | $\frac{1}{8}$ |
| $3^{-2}$ | $\frac{1}{9}$ |
| $5^{-1}$ | $\frac{1}{5}$ |
| $10^{-3}$ | $\frac{1}{1000}$ |