Indices can be negative numbers or fractions.

Fractional indices represent a root and negative indices represent a reciprocal:

• $a^{\frac{1}{y}}=\sqrt[y]{a}$
• $a^{\frac{x}{y}}=\sqrt[y]{a^x}$
• $a^{-x}=\dfrac{1}{a^x}$

Example 1

Simplify $a^{-2}$

$\dfrac{1}{a^{2}}$

Example 2

Simplify $b^{\frac{1}{3}}$

$\sqrt[3]{b}$

Example 3

Simplify $c^{-\frac{2}{5}}$

$\dfrac{1}{\sqrt[5]{c^{2}}}$ or $\dfrac{1}{(\sqrt[5]{c})^{2}}$

Example 4

Simplify $2^{-4}$

$\dfrac{1}{16}$

Example 5

Simplify $16^{\frac{3}{4}}$

$8$

Example 6

Simplify $27^{-\frac{2}{3}}$

$\dfrac{1}{9}$