N8NumberHigherGrades 6–9

Surds

Q: Why rationalise at all?

Convention and convenience: an answer like 63 is easier to compare and combine than 2 6, and exam questions ask for this form.

A surd is a square root that does not simplify to a whole number, such as $2$ or $35$ . Surds let you give answers exactly — a calculator can only ever show a rounded decimal. Higher papers use surds in Pythagoras, trigonometry, quadratic equations and "show that" proofs, so fluency here pays off across many topics.

45 min Video by Maths Genie AQA GCSE Maths

The big picture

Two rules do almost all the work: $ab = a \times b$ and $\frac{a}{b} = \frac{a}{b}$ . They let you pull square factors out of a root, and multiply and divide surds freely.

Treat a surd like an algebraic letter. $23 + 53 = 73$ just as $2 x + 5 x = 7 x$ , and $(2 + 3) (4 - 3)$ expands exactly like $(2 + x) (4 - x)$ — except that $3 \times 3 = 3$ .

SurdsCurated by Kepler, not produced by us. Watch it, then read the notes below.

Open on YouTube

What you'll learn

Simplify surds by finding square factors
Add, subtract, multiply and divide surds
Expand brackets containing surds
Rationalise the denominator of a fraction

Simplifying surds

Find the largest square number that is a factor of the number under the root. Split the root and take the square root of that factor.

Square numbers to look for: $4, 9, 16, 25, 36, 49, 64, 81, 100$ .

1The largest square factor of 72 is 36:

72 = 36 \times 2

72 = 36 \times 2 = 62

Answer

62

Your turnSame method, new numbers — try it before moving on.

Simplify $48$ .

Your answer

Type √ as sqrt, e.g. 5sqrt2.

Tip — Using a smaller square factor still works but needs another step: $72 = 218 = 2 \times 32 = 62$ .

Calculating with surds

Only like surds can be added or subtracted — simplify first to see whether they match. $2 + 3$ cannot be simplified.

Multiply the whole numbers together and the roots together: $32 \times 45 = 1210$ . Remember $a \times a = a$ .

50 = 52

18 = 32

8 = 22

52 + 32 - 22

Answer

62

Your turnSame method, new numbers — try it before moving on.

Simplify $75 - 12$ .

Your answer

Type √ as sqrt, e.g. 5sqrt2.

Expanding brackets

Expand exactly as in algebra, then simplify. Products such as $5 \times 5$ become whole numbers.

A useful special case: $(a + b) (a - b) = a^{2} - b$ . The surd terms cancel, leaving a whole number.

15 - 32 + 52 - 2

2Collect:

13 + 22

Answer

13 + 22

Your turnSame method, new numbers — try it before moving on.

Expand and simplify $(4 - 3)^{2}$ .

Your answer

Type it like 7 + 4sqrt3.

Rationalising the denominator

To remove a surd from a denominator, multiply the top and bottom by that surd: $\frac{a}{b} \times \frac{b}{b} = \frac{a b}{b}$ .

For a denominator like $3 + 2$ , multiply top and bottom by $3 - 2$ — the difference of two squares clears the surd.

1Multiply by

\frac{3}{3}

\frac{12 3}{3}

2Simplify:

43

Answer

43

Your turnSame method, new numbers — try it before moving on.

Rationalise the denominator of $\frac{10}{5}$ .

Your answer

Type √ as sqrt, e.g. 3sqrt2.

Your turnSame method, new numbers — try it before moving on.

Rationalise the denominator of $\frac{6}{4 - 10}$ .

Your answer

Type it like 3 + sqrt5.

Multiplying by 1

Rationalising does not change the value — you multiply by a fraction equal to 1. It simply puts the answer in a form that is easier to compare and to combine with other surds.

Think like an examiner

"Give your answer in the form $a b$ " means $b$ must be as small as possible: $218$ is not accepted where $62$ is expected.
Show the multiplication when rationalising, including the expanded denominator.
Exact answers mean surds: do not convert to decimals unless asked.

Surd rules

ab = a b, \frac{a}{b} = \frac{a}{b}

Split and combine roots.

\frac{a}{b} = \frac{a b}{b}

Rationalising a single surd.

(a + b) (a - b) = a^{2} - b

Clears the surd from a denominator.

Watch out for these

9 + 16 = 3 + 4

Roots do not split over addition:

25 = 5

2 + 3 = 5

Unlike surds cannot be combined.

(5)^{2} = 25

5 \times 5 = 5

Rationalising

\frac{1}{2 + 3}

by multiplying by

3

Multiply by

2 - 3

to use the difference of two squares.

Stretch yourself

A rectangle has sides $(3 + 2)$ cm and $(3 - 2)$ cm. Show that its diagonal is $22$ cm.

Hint — Use Pythagoras. Square each side carefully.

Questions students ask

9

a surd?

No — it equals 3 exactly. A surd is a root that is irrational, like

10

Why rationalise at all?

Convention and convenience: an answer like

\frac{6}{3}

is easier to compare and combine than

\frac{2}{6}

, and exam questions ask for this form.

Can I use a calculator for surds?

Many calculators simplify surds, which is useful for checking. But surd questions are usually on the non-calculator paper, so learn the methods.

Key takeaways

Simplify by taking out the largest square factor.
Add and subtract only like surds.
Multiply whole numbers and roots separately; $a \times a = a$ .
Expand brackets with surds like algebra.
Rationalise: multiply by $\frac{b}{b}$ , or by the conjugate $a - b$ .

Exam-style questions

Practise it the way it is examined

3 original questions · 8 marks, written to match the style and mark allocation of the real papers. Work on paper, then open the mark scheme and tick the marks you earned — method marks count even if the final answer slips.

Question 1Grade 63 marks

(a)Simplify $200$ .[1]
(b)Show that $27 + 48 = 73$ .[2]

Question 2Grade 72 marks

Expand and simplify $(5 + 23) (1 - 3)$ . Give your answer in the form $a + b 3$ .[2]

Question 3Grade 83 marks

Rationalise the denominator and simplify $\frac{2 + 7}{3 - 7}$ .[3]

Unlimited practice

Skill drill

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Surds

Simplifying surds

Calculating with surds

Expanding brackets

Rationalising the denominator

Practise it the way it is examined

Skill drill

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Related topics

Surds

Simplifying surds

Calculating with surds

Expanding brackets

Rationalising the denominator

Practise it the way it is examined

Skill drill

Practise this topic

Related topics