Pearson Edexcel International GCSE in Mathematics (Specification A) · 4MA1 · Free · no account needed

Surds

In one sentence. Exact answers with square roots: simplifying √48, expanding surd brackets, and rationalising denominators. Higher Tier.

Number — one of 8 modules in this strand, and one of 53 in Pearson Edexcel International GCSE Mathematics. Higher Tier only.

This module ends with Euclid. Learn Surds, teach it, then defend it against Euclid.

Where this sits in the specification

Specification 4MA1, 1.4 Powers and roots — surds (Higher). Incandio is aligned to this specification; it is not published by, endorsed by or affiliated with Pearson, and the sub-topic numbers are given so you can check every module against your own copy.

  • 1.4 — Powers and roots — surds (Higher)

1 · Understand it

What it is

A surd is a root left unevaluated because its decimal never ends: √3 exactly, rather than 1.732… approximately. Surd skills: simplify (√48 = 4√3), expand brackets, and rationalise — clearing roots from a denominator by multiplying top and bottom by a chosen surd.

Why it works

√48 = √(16 × 3) = √16 × √3 = 4√3 because roots split across multiplication. Rationalising works because √5 × √5 = 5 (kills a lone surd) and (3 − √3)(3 + √3) = 9 − 3 = 6 (the difference of two squares kills a mixed denominator — the cross terms cancel exactly).

When a question is asking for it

Whenever a question says 'give your answer in surd form' or 'exact value' — a decimal there usually loses the final accuracy mark even when numerically right, although correct working can still earn method marks. Surds appear inside Pythagoras, trigonometry with exact values, and quadratic formula answers.

How to hold it in your head

Treat √3 like a variable you can collect: 5√2 + 3√2 = 8√2 exactly as 5x + 3x = 8x. To simplify a surd, hunt for the biggest SQUARE factor hiding inside: 48 = 16 × 3, and the 16 escapes the root as 4.

2 · The mistakes that cost marks

  • √(a + b) = √a + √b. False: √(9+16) = 5, but √9 + √16 = 7. Roots split over MULTIPLICATION only.
  • Stopping early: √48 = 2√12 is true but unfinished — 12 still hides a square. Always extract the LARGEST square factor.
  • Rationalising by multiplying only the bottom. Multiply top AND bottom — you're multiplying by 1 in disguise, or you've changed the number.
  • For a denominator like 3 − √3, multiplying by 3 − √3 again. You need the CONJUGATE, 3 + √3, so the cross terms cancel.

3 · Worked examples

Laid out the way an examiner wants working shown — every step is a mark.

Expand and simplify (2 + √3)(5 − 2√3).

  1. All four products: 2 × 5 = 10, 2 × (−2√3) = −4√3, √3 × 5 = 5√3, √3 × (−2√3) = −2 × 3 = −6.
  2. Collect the whole numbers: 10 − 6 = 4. Collect the surds: −4√3 + 5√3 = √3.

Answer: 4 + √3

Rationalise the denominator of 10/√5.

  1. Multiply top and bottom by √5: (10√5)/(√5 × √5).
  2. √5 × √5 = 5, so this is 10√5/5.
  3. Simplify: 2√5.

Answer: 2√5

Rationalise the denominator of (1 + √2)/(3 − √2), giving your answer in the form (a + b√2)/c.

  1. Multiply top and bottom by the conjugate, 3 + √2.
  2. Denominator: (3 − √2)(3 + √2) = 9 − 2 = 7 — the cross terms cancel (difference of two squares).
  3. Numerator: (1 + √2)(3 + √2) = 3 + √2 + 3√2 + 2 = 5 + 4√2.
  4. Nothing cancels with the 7, so the answer is (5 + 4√2)/7.

Answer: (5 + 4√2)/7

4 · Prove it — the five questions

The five questions climb Grade 6 → 9+ and are marked inside Incandio on your own device, against authored accepted forms, with a diagnosis of the wrong answer you actually gave. The answers stay in the app so that the practice is worth doing; the questions themselves are here.

  1. Grade 6 — Simplify √48. (Type surds like 4√3 as 4sqrt3.)
  2. Grade 7 — Simplify √50 + √18.
  3. Grade 8 — Expand and simplify (3 + √2)(3 − √2).
  4. Grade 9 — Write 18/√6 + √24 in the form k√6, where k is an integer.
  5. 9+ — Rationalise the denominator and simplify: 6/(3 − √3).

Think you understand it?

Reading is the start. On Incandio a topic counts as mastered only once you have taught it and defended it — the two things a revision guide cannot ask of you.

Incandio this topic → Prove it. Teach it. Defend it. What does “Incandio it” mean?

  1. Prove it — five questions on Surds, marked on the spot, that tell you what was missing rather than just what was wrong.
  2. Teach it — Explain how Surds works — and why — to Ember, an AI pupil who does not have it yet. The method matters more than the vocabulary.
  3. Defend it — Argue Surds against Euclid — three rounds, marked against descriptors you can read before you begin.
  4. Then the exam — read all 10 authored exam-style questions on this unit, each with its command word and its tariff, then have them marked on Incandio with the mistake you actually made named.

Completely free. No subscriptions, no paywalls, no account needed to start.

Carry on through the course