Why does any nonzero number to the zero power equal 1?

Let $a e0$ and choose a positive integer $n$. A nonzero number divided by itself equals $1$, so rac{a^n}{a^n}=1. The quotient rule for powers with the same base also says rac{a^n}{a^n}=a^{n-n}=a^0. Both expressions describe the same quotient, so $a^0=1.$ This is not an isolated convention: it is the value that keeps the exponent rules consistent.

Take $a=5$ and $n=3$. On one hand, rac{5^3}{5^3}=rac{125}{125}=1. On the other, subtracting exponents gives rac{5^3}{5^3}=5^{3-3}=5^0. Therefore $5^0=1$. The same check works with a negative base: $(-2)^4/(-2)^4=16/16=1=(-2)^0$.

For a nonzero base, lowering the exponent by $1$ means dividing by the base. With $a=3$, $3^3=27,quad 3^2=9,quad 3^1=3,quad 3^0=1.$ Each term is the previous one divided by $3$. Continuing gives $3^{-1}=1/3$. The value $1$ at exponent zero therefore connects positive and negative exponents without breaking the pattern.

The derivation uses $a^n/a^n$, which requires a nonzero denominator. If $a=0$ and $n>0$, then $0^n/0^n=0/0$, which is undefined. That is why the standard rule is stated as $a^0=1$ for $a e0$.

The expression $0^0$ is sometimes assigned the value $1$ in discrete settings such as certain combinatorial formulas, while in analysis it can appear as an indeterminate form in a limit. The context must be stated: the derivation above does not establish a universal value for $0^0$.