3.4 Negative and fractional indices — Learn

Prerequisites: roots and integer index laws; fractions and reciprocals.

Objectives: interpret fractional indices as roots; combine roots and powers; evaluate negative fractional indices.

By the end of this section, you can interpret and calculate with fractional and negative fractional indices.

This section covers the Higher content in N7. A fractional index connects powers with roots.

Unit fractions as roots

For a positive number $a$, the index law shows that squaring $a^{1/2}$ gives $a$. So $a^{1/2}=\sqrt a$. Also, $a^{1/3}=\sqrt[3]{a}$.

Example 1. Work out $64^{1/3}$.

Other fractional indices

The denominator gives the root. The numerator gives the power. For example, $a^{2/3}$ means take the cube root, then square.

Example 2. Work out $81^{3/4}$.

Negative fractional indices

A negative index means take the reciprocal. So $a^{-m/n}=1/a^{m/n}$ when $a$ is positive and non-zero.

Example 3. Work out $32^{-2/5}$.

Watch out. The denominator tells you the root and the numerator tells you the power. In $a^{2/3}$, taking the square root would swap these roles. Check by rewriting the expression with a radical.

You should now be able to

• rewrite a unit fractional index as a root;
• interpret $a^{m/n}$ as an nth root raised to the mth power;
• use a negative index to form a reciprocal;
• evaluate positive numbers with negative fractional indices.