The laws of indices let us make sense of a power even when the index is zero. Take any non-zero number and divide it by itself — the answer is always 1. Using the division law, dividing a^n by a^n gives a^{n-n} = a^0, so:

This holds for every non-zero base: 5^0 = 1, 100^0 = 1, even (0.5)^0 = 1. The index 0 says “no copies of the base” — and an empty product is 1, not 0.

Letting the index drop below zero keeps the same pattern. Each time the index falls by one, we divide by the base once more — so a negative index means “one over” the matching positive power:

For example, 2^{-3} = \frac{1}{2^3} = \frac{1}{8} = 0.125. The minus sign in the index does not make the number negative — it flips it into a fraction.