Surds: brackets and harder denominators

Surds give exact answers. You can expand brackets that contain them and clear them from the bottom of a fraction.

GCSE · 1MA1 · N8, A4
International GCSE · 4MA1 · 1.4
About 18 minutes
Higher tier

Multiplying surds

√a × √b = √(ab), and √a × √a = a.
3√2 × 5√6 = 15√12 = 15 × 2√3 = 30√3
(√7)² = 7

Expanding brackets

Multiply every term by every term, just like algebra.
(2 + √3)(4 − √3) = 8 − 2√3 + 4√3 − 3 = 5 + 2√3
(3 + √2)² = 9 + 3√2 + 3√2 + 2 = 11 + 6√2
(√5 + 1)(√5 − 1) = 5 − 1 = 4: the surds cancel.

Rationalising (a ± √b)

Multiply top and bottom by the same bracket with the sign changed.
4(3 − √5) × (3 + √5)(3 + √5) = 4(3 + √5)(9 − 5) = 4(3 + √5)4 = 3 + √5
This works because (a − √b)(a + √b) = a² − b has no surd.

Worked example

Expand and simplify (5 − √3)². Write it in the form a + b√3.

(5 − √3)(5 − √3)
= 25 − 5√3 − 5√3 + 3
= 28 − 10√3

Answer: 28 − 10√3

Key idea

Expand surd brackets like algebra, using √a × √a = a. To rationalise a + √b on the bottom, multiply top and bottom by a − √b.

Key words

Surd: A root that cannot be written exactly as a whole number or fraction, such as √5, left in exact form.
Term: A single number, letter or product of them in an expression, such as 5x or −3.

All GCSE and International GCSE Mathematics key terms and definitions →

Common mistakes

Not “5 − 2√3”. Check the sign of −2√3 + 4√3.
Not “11 + 2√3”. √3 × √3 = 3, and it is subtracted.
Not “9 + 2”. You must include the two middle terms.
Not “11”. The middle terms do not cancel here.

Quick check

Try two quick questions

See if it has sunk in. Answers are marked straight away.

Extra practice

2 more questions

Try each one, then tap to check.

Expand and simplify (3 + √2)(5 − √2)
1. 13 − 2√2
2. 17 + 2√2
3. 13
4. 13 + 2√2
Show the answer
Answer: 13 + 2√2
15 − 3√2 + 5√2 − 2 = 13 + 2√2.
Expand and simplify (5 + √3)²
1. 28
2. 25 + 10√3
3. 28 + 5√3
4. 28 + 10√3
Show the answer
Answer: 28 + 10√3
(5 + √3)(5 + √3) = 25 + 5√3 + 5√3 + 3 = 28 + 10√3.

Check you have got it

Answer 6 quick questions with instant marking. If you get one wrong, GCSE-ready shows you why and gives you another go. It is free, and you do not need an account.

Practise this lesson free (6 questions)All GCSE and International GCSE Mathematics lessons

Video for this lesson coming soon. Subscribe on YouTube

Sources and references

Checked by the GCSE-ready editorial team, 17 October 2026. Last updated 2 October 2026. Spotted a mistake? Tell us and we will fix it.