Fractional and Negative Indices

When indices are fractions or negative there are special rules you will need to learn to deal with them.

Fractional Indices

A fractional index represents a root. The general rule is:

$$a^{\frac{m}{n}}=\sqrt[n]{a^m}={\left(\sqrt[n]{a}\right)}^m$$

We can use this rule:

$${16}^{\frac{1}{2}}=\sqrt{16}=4$$

$${27}^{\frac{2}{3}}={\left(\sqrt[3]{27}\right)}^2=3^2=9$$

Negative Indices

A negative index represents the reciprocal of the base raised to the corresponding positive index. The rule is:

$$a^{-n}=\frac{1}{a^n}$$

We can use this rule:

$$5^{-2}=\frac{1}{5^2}=\frac{1}{25}$$

Combining Fractional and Negative Indices

When you have both fractional and negative indices, apply both rules:

$$a^{-\frac{m}{n}}=\frac{1}{a^{\frac{m}{n}}}=\frac{1}{\sqrt[n]{a^m}}$$

We can use this rule:

$$8^{-\frac{2}{3}}=\frac{1}{8^{\frac{2}{3}}}=\frac{1}{{\left(\sqrt[3]{8}\right)}^2}=\frac{1}{4}$$