Solving exponential equations with same base

The zero exponent rule states that any non-zero base raised to the power of zero equals 1.

$a^0=1$ for any $a\ne 0$.

Here $a=5$, so $5^0=1$.

The answer is 1.

$5^0\ne 0$ because the zero exponent rule does not produce 0 — that is a common confusion with the base being 0.

$5^0\ne 5$ because $5^1=5$, not $5^0$.

$5^0$ is perfectly defined — the base is non-zero, so the rule applies.

Only 1 remains, which matches the zero exponent rule $a^0=1$.

The negative exponent rule states $a^{-n}=\frac{1}{a^n}$.

$2^{-3}=\frac{1}{2^3}$.

$2^3=8$, so $2^{-3}=\frac{1}{8}$.

The answer is $\frac{1}{8}$, a small positive fraction.

A negative exponent does NOT make the result negative. $2^{-3}\ne -8$.

Again, negative exponents produce positive fractions, not negative fractions. Eliminated.

$8=2^3$, which would be the result of a positive exponent, not a negative one.

Only $\frac{1}{8}$ remains, correctly applying $a^{-n}=\frac{1}{a^n}$.