Indices summary

Key learning points

• $a^m a^n=a^{m+n}$, $(a^m)^n=a^{mn}$, $\frac{a^m}{a^n}=a^{m-n}$.
• $a^0=1$ ($a\ne0$); $a^{-n}=\frac{1}{a^n}$.
• $a^{1/n}=\sqrt[n]{a}$; $a^{m/n}=(\sqrt[n]{a})^m$.
• Use the same base before combining indices.

Worked example 1 — Simplify

Simplify $a^3\times a^4$.

Working

Add indices.

Worked example 2 — Division

Simplify $\frac{x^8}{x^3}$.

Working

Subtract indices.

Worked example 3 — Negative index

Write $2^{-3}$ as a fraction.

Working

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

Scaffolded Learning Sheet and answers (PDFs)

This page is a short, searchable overview for revision and search engines. Use the Scaffolded Learning Sheet first, then the Answers PDF for a full written solution. The printable PDFs remain the main classroom resource.