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:

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