GCSE & IGCSE maths practice

Surds questions and answers

A surd is a root that isn't a whole number, like $\sqrt{2}$ or $\sqrt{50}$ . Surd answers are exact, which is why exam questions ask for them. The questions start with simplifying a single surd, then adding and multiplying, expanding brackets, and finish with rationalising denominators. Type a surd as sqrt(2) or √2.

11 questions · Higher tier · Every answer is checked as you type it, with a full worked solution. Updated 26 Oct 2026.

Start the questions Laws of indices: lesson and examples Square root simplifier

Where surds comes up

GCSE (AQA, Edexcel, OCR)	Higher only: calculating exactly with surds, simplifying surd expressions and rationalising denominators.
Cambridge IGCSE (0580)	Extended: understanding and using surds, including simplifying and rationalising denominators.
Edexcel International GCSE (4MA1)	Higher: understanding and manipulating surds, including rationalising a denominator.

The method

To simplify √n, find the largest square number that divides n: √50 = √25 × √2 = 5√2.
You can only add or subtract like surds (the same number under the root): 2√3 + 3√3 = 5√3.
√a × √b = √(ab), and √a × √a = a.
Expand brackets with surds exactly like algebra.
To rationalise 1/√a, multiply top and bottom by √a. For 1/(a − √b), multiply top and bottom by (a + √b).

Surds questions

Try each one before looking at the hint. If you're stuck, the AI tutor can talk you through it step by step.

Warm-up

Question 1Warm-upHigher

Simplify $\sqrt{50}$

Question 2Warm-upHigher

Simplify $\sqrt{12} + \sqrt{27}$

Question 3Warm-upHigher

Simplify $\sqrt{8} \times \sqrt{6}$

Exam-style

Question 4Exam-styleHigher

Expand and simplify $(3 + \sqrt{2})(3 - \sqrt{2})$

Question 5Exam-styleHigher

Expand and simplify $(2 + \sqrt{3})^2$ . Give your answer in the form $a + b\sqrt{3}$ .

Question 6Exam-styleHigher

Rationalise the denominator of $\dfrac{6}{\sqrt{3}}$ , simplifying your answer.

Question 7Exam-styleHigher

Rationalise the denominator of $\dfrac{5}{2\sqrt{5}}$ , simplifying your answer.

Question 8Exam-styleHigher

$\sqrt{72} - \sqrt{32}$ can be written as $a\sqrt{2}$ . Find $a$ .

Question 9Exam-styleHigher

A rectangle has sides of length $(\sqrt{3} + 1)$ cm and $(\sqrt{3} - 1)$ cm. Find its area.

Challenge

Question 10ChallengeHigher

Rationalise the denominator of $\dfrac{4}{3 - \sqrt{5}}$ , simplifying your answer.

Question 11ChallengeHigher

Write $\dfrac{\sqrt{5} + 1}{\sqrt{5} - 1}$ in the form $a + b\sqrt{5}$ , where $a$ and $b$ are rational.

a =b =

Common mistakes

Adding unlike surds: √2 + √3 is not √5.
Simplifying with a square factor that isn't the largest: √72 = 6√2, not 3√8.
Forgetting the middle terms when squaring: (2 + √3)² = 7 + 4√3, not 4 + 3.
Multiplying only the bottom when rationalising.

Stuck on one of these?

The TutorAura AI tutor works through it with you step by step, with the reason for each step, then gives you one like it to try. It checks your working too.

Try the AI tutor free

Sources

The questions are our own. The “where it comes up” notes are from each board's published specification, checked on 26 October 2026:

Practise maths with your own AI tutor

Questions at your level, help when you're stuck, and a plan for your exams. Free for 3 days, no card needed.

Start free trial