Zero and negative indices

For \(a\ne0\), cancellation gives \(\dfrac{a^m}{a^m}=1\), while the index law gives \(a^{m-m}=a^0\). Thus \(a^0=1\). Continuing the same division pattern gives \(a^{-n}=\dfrac1{a^n}\). A negative index means a reciprocal.

Example

\(5^0=1\), \(2^{-3}=\dfrac18\), and \(\left(\dfrac23\right)^{-2}=\left(\dfrac32\right)^2=\dfrac94\).

\(\begin{aligned}\dfrac{3x^{-2}y^3}{6xy^{-1}}&=\dfrac12x^{-3}y^4\\&=\dfrac{y^4}{2x^3}\end{aligned}\), where \(x\ne0\) and \(y\ne0\).

Zero cannot be the base of a negative power because the reciprocal would divide by zero. The rule \(a^0=1\) also requires \(a\ne0\).