The index laws are the rules by which indices (powers) may be combined.
The basic rules are: \[\begin{align*} a^1 &= a \\ a^m \cdot a^n &= a^{m+n} \\ (a^m)^n &= a^{mn} \\ a^m \cdot b^m &= (ab)^m \end{align*}\] From these we can derive other important rules: \[\begin{align*} a^0 &= 1 \\ \frac{a^m}{a^n} &= a^{m-n} \\ a^{-m} &= \frac{1}{a^m} \\ a^{1/n} &= \sqrt[n]{a} \end{align*}\]These rules are valid for \(a\) and \(b\) positive real numbers and \(m\) and \(n\) rational numbers, or \(a\) and \(b\) any numbers and \(m\) and \(n\) integers.
