Learn4024Paper 2 · CalculatorNumberSurds

Number · Subtopic 6 of 8

Surds

A surd is a root that can't be simplified to a whole number, like $\sqrt2$ . Keep surds exact: pull out square factors ( $\sqrt{48}=4\sqrt3$ ), collect like surds, and remove surds from denominators by multiplying top and bottom by the right surd or by the conjugate.

Cambridge 4024 Paper 1Cambridge 4024 Paper 2Edexcel 4MA1 Papers 1H, 2HAQA 8300 Papers 1–3≈ 30 minRevision 5 · updated 2026-10-27

Section 1 of 14

Learning objectives

Simplify surds using square factors.
Add, subtract, multiply and expand surds.
Rationalise denominators of the form $\sqrt b$ .
Rationalise denominators of the form $a\pm\sqrt b$ .

Pause and think

Your calculator says $\sqrt2=1.414213562\ldots$ . Why might an exam ask you to leave the answer as $\sqrt2$ instead?

Section 2 of 14

Core idea

Start hereExact, not approximate

$\sqrt2=1.41421356\ldots$ never ends or repeats. Write $1.414$ and you've introduced an error that grows through a calculation. Keeping the root as a symbol, a surd, keeps the answer exact.

The big idea

$\sqrt{ab}=\sqrt a\sqrt b$ : pull out square factors. Rationalise: $\frac a{\sqrt b}\times\frac{\sqrt b}{\sqrt b}$ ; for $p\pm\sqrt q$ multiply by the conjugate.

1. Simplifying

The diagonal of a 1 by 1 square is √2

The decimal never ends or repeats. Writing

\sqrt2

is exact;

1.41

is only an approximation.

Worked exampleSimplify

$\sqrt{72}$ ; $\sqrt{12}+\sqrt{27}$ ; $(2+\sqrt3)(1-\sqrt3)$ .

Goal: Simplest surd form.
Strategy: Largest square factor; expand brackets fully.

$\sqrt{36\times2}=6\sqrt2$
36 is the largest square factor.
$2\sqrt3+3\sqrt3=5\sqrt3$
Like surds add like $x$ 's.
$2-2\sqrt3+\sqrt3-3=-1-\sqrt3$
$\sqrt3\times\sqrt3=3$ .

Answer

6\sqrt2

5\sqrt3

-1-\sqrt3

✓ Check

\sqrt{72}=8.485=6\times1.414

✓.

2. Rationalising

Why the conjugate works

$(\sqrt3-1)(\sqrt3+1)$
Multiply by the "opposite sign" version.
$=3+\sqrt3-\sqrt3-1=2$
Middle terms cancel: difference of squares.

Worked exampleRationalise

$\frac{10}{\sqrt5}$ and $\frac2{\sqrt3-1}$ .

Goal: No surd in the denominator.
Strategy: Multiply top and bottom.

$\frac{10\sqrt5}{5}=2\sqrt5$
By $\sqrt5$ .
$\frac{2(\sqrt3+1)}{2}=\sqrt3+1$
By the conjugate.

Answer

2\sqrt5

;

\sqrt3+1

✓ Check

10/2.236=4.472=2\times2.236

✓.

Common misconception $\sqrt{a}+\sqrt b\neq\sqrt{a+b}$

\sqrt9+\sqrt{16}=7

, but

\sqrt{25}=5

Pause and think

Simplify $\sqrt5\times\sqrt{20}$ .

Summary

Largest square factor.
Conjugate to rationalise.

Surd: Irrational root in exact form.
Rationalise: Remove roots from the denominator.

HistoryWhy "surd"? The word comes from the Latin surdus, "deaf" or "mute", a translation of an Arabic term used by al-Khwarizmi for numbers that couldn't be "spoken" as ratios of whole numbers.

Fun factPaper sizes. A-series paper has sides in the ratio

1:\sqrt2

, so folding a sheet in half gives the next size down with exactly the same shape. An A4 sheet is

210

mm by

297

mm, and

297\div210\approx1.414\approx\sqrt2

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