Surds summary

Key learning points

• $\sqrt{ab}=\sqrt{a}\sqrt{b}$ for $a,b\ge0$; e.g. $\sqrt{12}=2\sqrt{3}$.
• Rationalise single-surd denominators by multiplying by the surd; use conjugates for binomials.
• Collect like surds as like terms.
• Leave answers in simplest surd form unless asked for decimals.

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

Worked example 4 — Simplify

Simplify $\sqrt{50}+\sqrt{18}$.

Working

Extract square factors, then collect.

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.