Indices and Surds Topic Review summary

Key learning points

• $a^m a^n=a^{m+n}$, $\frac{a^m}{a^n}=a^{m-n}$, $(a^m)^n=a^{mn}$.
• $a^{-n}=\frac{1}{a^n}$, $a^{1/n}=\sqrt[n]{a}$, $a^{m/n}=\sqrt[n]{a^m}$.
• Surds: $\sqrt{ab}=\sqrt{a}\sqrt{b}$; simplify $\sqrt{12}=2\sqrt{3}$.
• Rationalise denominators (multiply by conjugates when needed).
• Combine index and surd skills in single exam questions.

Worked example 1 — Index

Simplify $8^{2/3}$.

Working

$(2^3)^{2/3}=2^2$.

Worked example 2 — Rationalise

Rationalise $\frac{1}{\sqrt{3}}$.

Working

Multiply by $\sqrt{3}/\sqrt{3}$.

Worked example 3 — a^m a^n=a^{m+n}, \a^m{a^n}=a^{m-n}, (a^m)^

$a^m a^n=a^{m+n}$, $\\frac{a^m}{a^n}=a^{m-n}$, $(a^m)^n=a^{mn}$.

Working

Attempt without notes, then check the answers PDF.

Worked example 4 — a^{-n}=\1{a^n}, a^{1/n}=\\sqrt[n]{a}, a^{m

$a^{-n}=\\frac{1}{a^n}$, $a^{1/n}=\\sqrt[n]{a}$, $a^{m/n}=\\sqrt[n]{a^m}$.

Working

Attempt without notes, then check the answers PDF.

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.