Fractional & negative indices – GCSE Maths

Higher · Grade 6 · Non-calculator

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A negative index is a reciprocal: $a^{-n}=\frac{1}{a^n}$, so $2^{-3}=\frac{1}{8}$. A fractional index is a root: $a^{\frac{1}{2}}=\sqrt{a}$, $a^{\frac{1}{3}}=\sqrt[3]{a}$, and $a^{\frac{m}{n}}={\left(\sqrt[n]{a}\right)}^m$ (root first, usually, to keep numbers small). Combine with the ordinary index laws when the bases match.

$16^{\frac{3}{4}}=(\sqrt[4]{16}{)}^3=2^3=8$. $27^{-\frac{2}{3}}=\frac{1}{(\sqrt[3]{27}{)}^2}=\frac{1}{9}$. Write the root and the power as two separate steps so a sign or a cube-root slip is visible. $a^0=1$ still holds.

Negative means reciprocal. Unit fraction means root. $\frac{m}{n}$ means root $n$ first, then power $m$. Flip a fraction base when the index is negative. Do not do $27^{2/3}$ as $729$ then cube-root — root first.

Objectives

• Work out a fractional index by finding the root first.
• Work out a negative integer power as a unit fraction.
• Combine a negative sign with a fractional index.

Key points

• A negative index means reciprocal: $a^{-n}=\frac{1}{a^n}$.
• A unit fraction index is a root: $a^{\frac{1}{n}}=\sqrt[n]{a}$.
• A fraction like $\frac{m}{n}$ means root first, then power: $a^{\frac{m}{n}}=(\sqrt[n]{a}{)}^m$.
• Flip a fraction base when the index is negative: ${\left(\frac{a}{b}\right)}^{-n}={\left(\frac{b}{a}\right)}^n$.

Worked examples

Questions and answers

GCSE Maths fractional & negative indices questions and answers. Try each one, then open the answer. Easy questions come first.

Common mistakes

Most lost marks on this topic are one slip.

1. Squaring the base and ignoring the root

   Take the root named by the denominator first

2. Writing a negative number for a negative index

   Write the reciprocal instead

3. Forgetting the reciprocal after a negative fractional index

   Flip the positive result at the end

Exam tips

• Be ready to do this without a calculator. Paper 1 is the non-calculator paper.
• If the question says “show that”, the result is given. The marks are for the steps that reach it.
• The method is the same on AQA-style and Edexcel-style papers.
• Write the intermediate line (e.g. $\sqrt[3]{27}=3$) — it earns a method mark even if you then miscount a power.
• Learn the small powers of 2, 3 and 5 by heart; these questions are set on a non-calculator paper.

Printable worksheet

The same questions are in a PDF you can print. Answers start on a later page.

Download worksheet (PDF)

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Questions students ask

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More on Fractional & negative indices

• Fractional & negative indices worked examples and method
  The method in order, key facts and the full lesson explanation

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Last updated 9 November 2026.