Negative and fractional indices

In a nutshell

Indices are variables raised to a power. The laws of indices show how to manipulate expressions involving powers, including those which have negative or fractional powers.

Negative indices

When an index has a negative power, this means that it is one divided by the index raised to the positive of the power. This is also called the reciprocal of the index. You can use this rule:

$\boxed{a^{-n} = \dfrac{1}{a^n}}$

Example 1

What is $4^{-2}$ ?

$\begin{aligned}4^{-2}&= \dfrac{1}{4^2}\\\\&=\underline{\dfrac{1}{16}}\end{aligned}$ 

Fractional indices

Fractional indices are indices whose powers are expressed in fractions. The denominator of the fraction refers to the root of the index. The answer is then raised to the power, which is the value of the numerator.

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

Common indices

Some of the most commonly used indices are as follows:

Reciprocal of $a$	$\dfrac{1}{a}$	$a^{-1}$
Square root of $a$	$\sqrt [2]{a}$	$a^\frac{1}{2}$
Cube root of $a$	$\sqrt [3]{a}$	$a^{\frac{1}{3}\\}$

Example 2

What is $\left(\dfrac{36}{49}\right)^\frac12$ ?

$\begin{aligned}\left(\dfrac{36}{49}\right)^\frac12&= \sqrt {\dfrac{36}{49}}\\\\&=\underline{\dfrac{6}{7}}\end{aligned}$

Example 3

What is $16^{-\frac{3}{2}}$ ?

$\begin{aligned}16^{-\frac{3}{2}}&=\frac{1}{16^\frac32}\\\\&=\frac1{16^{\frac12\times 3}}\\\\&=\frac1{(\sqrt{16})^3}\\\\&=\frac1{4^3}\\\\&=\underline{\frac1{64}}\end{aligned}$