How Fractional and Negative Indices Work

📋 Spec coverage: GCSE Edexcel Mathematics (1MA1) This article covers the negative-index aspect of Edexcel N7: calculating with integer indices (Foundation and Higher), and Edexcel N7 (Higher): calculating with fractional indices.

These skills may be assessed on any of the three GCSE Maths papers.

Negative indices mean reciprocals, while fractional indices mean roots. If an index is both negative and fractional, apply both rules carefully.

Negative indices

A reciprocal is found by writing a non-zero number as the denominator of a fraction with numerator $1$. For any non-zero value of $a$:

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

For example:

$$2^{-3}=\frac{1}{2^3}=\frac{1}{8}$$

This follows from the division law for indices:

$$\frac{a^2}{a^5}=a^{2-5}=a^{-3}$$

Cancelling the same fraction also gives:

$$\frac{a^2}{a^5}=\frac{1}{a^3}$$

Therefore, $a^{-3}=\frac{1}{a^3}$.

Fractional indices

The denominator of a fractional index tells you which root to use:

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

The numerator tells you the power:

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

For example:

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

You can often find the root first because this keeps the arithmetic simpler. Here, cube-rooting $8$ before squaring avoids working with a larger number.

Worked example

Calculate $27^{-\frac{2}{3}}$.

First, use the negative index to take the reciprocal:

$$27^{-\frac{2}{3}}=\frac{1}{27^{\frac{2}{3}}}$$

Then use the denominator $3$ as the root and the numerator $2$ as the power:

$$27^{-\frac{2}{3}}=\frac{1}{{\left(\sqrt[3]{27}\right)}^2}=\frac{1}{3^2}=\frac{1}{9}$$

A quick check helps: the original base is positive, so the answer must be positive. The negative index also means the result should be a reciprocal.

An efficient order is: deal with the negative sign by taking the reciprocal, evaluate the root from the denominator, then apply the numerator as a power. Write each stage separately so your method is clear.

Common exam mistake: a negative index does not make the value negative. For example, $4^{-1}$ is $\frac{1}{4}$, not $-4$. Also, do not swap the numerator and denominator of a fractional index: the denominator is always the root.

Completeness check: The negative-index rule, fractional-index rule, their connection to reciprocals and roots, and calculations combining both rules are covered at the required GCSE depth.