Recurring Decimals and Surds summary

Key learning points

• Let $x=0.\dot{a}\dot{b}\ldots$, multiply by $10^k$, subtract to clear the repeat.
• Simplify $\sqrt{a}$: factor out square factors.
• Rationalise denominators: multiply by a suitable surd conjugate when needed.

Worked example 1 — Recurring decimal

Express $0.\dot{3}$ as a fraction.

Working

Let $x=0.\dot{3}$; $10x-x=3$.

Worked example 2 — Simplify surd

Simplify $\sqrt{50}$.

Working

$\sqrt{25\times2}$.

Worked example 3 — Rationalise

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

Working

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

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.