Edexcel 4MA1Cambridge 0580
IGCSE Math Revision Questions by topic Revision notes Past papers Student hub

Cambridge IGCSE Maths (0580) · Topic 1: Number · 1.18

Cambridge IGCSE Maths 0580 Surds Questions

Surds questions with worked answers for Cambridge IGCSE Mathematics (0580). Each question says whether it is Core or Extended and whether it is for a calculator or non-calculator paper. Try each one before you open the answer.

Practise surds →Surds notes

What the syllabus asks

Summarised in our own words: check the exact wording in the official 0580 syllabus.

Surds questions: Extended

Extended content, for Papers 2 and 4.

Question 1 Extended non-calculator · easy

Simplify
(a) \(\sqrt{50}\)
(b) \(\sqrt{48}\)

Show the worked answer
  1. (a) \(\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}\)
  2. (b) \(\sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3}\)

Answer: (a) \(5\sqrt{2}\) (b) \(4\sqrt{3}\)

Question 2 Extended non-calculator · easy

Write \(\sqrt{20} + \sqrt{45}\) as a single surd in its simplest form.

Show the worked answer
  1. \(\sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}\) and \(\sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5}\).
  2. \(2\sqrt{5} + 3\sqrt{5} = 5\sqrt{5}\)

Answer: \(5\sqrt{5}\)

Question 3 Extended non-calculator · medium

Simplify \(\sqrt{98} - \sqrt{32}\).

Show the worked answer
  1. \(\sqrt{98} = \sqrt{49 \times 2} = 7\sqrt{2}\) and \(\sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2}\).
  2. \(7\sqrt{2} - 4\sqrt{2} = 3\sqrt{2}\)

Answer: \(3\sqrt{2}\)

Question 4 Extended non-calculator · medium

Multiply out \((5 + \sqrt{3})(5 - \sqrt{3})\) and simplify your answer.

Show the worked answer
  1. \(25 - 5\sqrt{3} + 5\sqrt{3} - 3\)
  2. \(= 22\)

Answer: \(22\)

Question 5 Extended non-calculator · medium

Expand and simplify \((2 + \sqrt{5})^2\).

Show the worked answer
  1. \((2 + \sqrt{5})(2 + \sqrt{5}) = 4 + 2\sqrt{5} + 2\sqrt{5} + 5\)
  2. \(= 9 + 4\sqrt{5}\)

Answer: \(9 + 4\sqrt{5}\)

Question 6 Extended non-calculator · medium

Rationalise the denominator of \(\frac{6}{\sqrt{3}}\). Simplify your answer.

Show the worked answer
  1. Multiply top and bottom by \(\sqrt{3}\): \(\frac{6\sqrt{3}}{3}\).
  2. \(= 2\sqrt{3}\)

Answer: \(2\sqrt{3}\)

Question 7 Extended non-calculator · hard

Rationalise the denominator of \(\frac{4}{3 - \sqrt{5}}\). Simplify your answer.

Show the worked answer
  1. Multiply top and bottom by \(3 + \sqrt{5}\).
  2. Bottom: \((3 - \sqrt{5})(3 + \sqrt{5}) = 9 - 5 = 4\).
  3. \(\frac{4(3 + \sqrt{5})}{4} = 3 + \sqrt{5}\)

Answer: \(3 + \sqrt{5}\)

Question 8 Extended non-calculator · hard

A rectangle has sides of length \((\sqrt{8} + 1)\) cm and \((\sqrt{2} - 1)\) cm.
Find its area in the form \(a + b\sqrt{2}\), where \(a\) and \(b\) are integers.

Show the worked answer
  1. \((\sqrt{8} + 1)(\sqrt{2} - 1) = \sqrt{16} - \sqrt{8} + \sqrt{2} - 1\)
  2. \(= 4 - 2\sqrt{2} + \sqrt{2} - 1 = 3 - \sqrt{2}\)

Answer: \(3 - \sqrt{2}\) cm²

Every question on this page is original, written for IGCSE Math Revision rather than taken from Cambridge papers, and each answer was re-solved independently before publishing. Questions with a mark scheme come from our 0580 practice bank (M method, A accuracy, B independent marks).

Keep going

Questions students ask

Which Cambridge 0580 papers can test surds?

Surds is Extended content (syllabus 1.18), so it is tested only on the Extended papers, Paper 2 (non-calculator) and Paper 4 (calculator). Core candidates do not need it.

Are these Cambridge past-paper questions?

No. Every question here was written for IGCSE Math Revision in the style of the 0580 papers, and each answer was checked independently before publishing. For real exam practice, use the Cambridge past papers linked from our 0580 past papers page.

Where can I practise more surds questions?

In the 0580 practice for topic 1 (Number), where every question is marked and tagged Core or Extended. Practice is part of the IGCSE plan, with a free preview; the questions on this page are free.

More number questions