Negative and fractional indices

The idea

A negative power means a reciprocal: $x^{-n} = \frac{1}{x^n}$. A fractional power means a root: $x^{\frac{1}{n}} = \sqrt[n]{x}$ and $x^{\frac{m}{n}} = (\sqrt[n]{x})^m$.

1. Deal with a negative sign first by taking the reciprocal.
2. Then the bottom of the fraction: take that root.
3. Then the top: raise to that power. $8^{-\frac{2}{3}} = \frac{1}{(\sqrt[3]{8})^2} = \frac{1}{4}$.

Watch out A negative power does not make the answer negative: $2^{-3} = \frac{1}{8}$.