Zero and Negative Exponents

Two special exponents extend the laws you already know.

Zero exponent. For any non-zero base, $a^0 = 1$. Why? By the quotient law, $\dfrac{a^m}{a^m} = a^{m-m} = a^0$. But any non-zero number divided by itself is $1$. So $a^0$ must equal $1$.

Negative exponent. A negative exponent means a reciprocal:

Why? Follow the pattern down: $2^3 = 8,\; 2^2 = 4,\; 2^1 = 2,\; 2^0 = 1$ — each step halves. Keep halving: $2^{-1} = \tfrac{1}{2},\; 2^{-2} = \tfrac{1}{4}$. A negative exponent just continues the pattern past zero.

Worked example. $5^0 = 1$, $\; 2^{-3} = \dfrac{1}{2^3} = \dfrac{1}{8}$, and $\dfrac{1}{x^{-2}} = x^2$ (a negative exponent in the denominator flips up top).