What is important to know about Surds?

√(ab) = √a × √b — use this to simplify surds.

What is important to know about Surds?

To simplify √n, find the largest perfect square factor of n.

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Number · Higher

Surds

Surds are irrational numbers written as square roots that cannot be simplified to exact integers. You need to simplify surds, expand brackets containing surds, and rationalise denominators.

Revising surds the night before an exam? These are condensed revision notes — the facts, formulas and mistakes examiners actually mark, with no long explanations. Prefer a full walkthrough with step-by-step reasoning instead? Search “Surds GCSE” on the site for the complete guide.

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Key facts to remember

1√a × √a = a
2√(ab) = √a × √b — use this to simplify surds.
3To simplify √n, find the largest perfect square factor of n.
4Rationalise the denominator: multiply numerator and denominator by the surd in the denominator.
5For (a + √b)(a − √b) = a² − b — called the difference of two squares.
6Surds are exact values; leaving an answer in surd form is more accurate than a decimal.

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Formulas

Simplify

√(a²b) = a√b

Rationalise

a/√b = a√b / b

Multiply top and bottom by √b

Conjugate pair

(p + √q)(p − √q) = p² − q

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Worked examples

Example 1

Simplify √72

Working

Find the largest perfect square factor of 72: 36 × 2 = 72
√72 = √(36 × 2) = √36 × √2
= 6√2

Answer6√2

Example 2

Rationalise the denominator of 5/√3

Working

Multiply numerator and denominator by √3
5/√3 × √3/√3 = 5√3 / (√3 × √3)
= 5√3 / 3

Answer5√3 / 3

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Common mistakes

✗√(a + b) ≠ √a + √b — you cannot split a sum under a square root.

✗Not finding the largest perfect square factor (e.g. simplifying √72 as 2√18 instead of 6√2).

✗Forgetting to rationalise when the question asks for an exact answer with a rational denominator.

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Exam tips

✓Always check if your simplified surd can be simplified further — look for perfect square factors.

✓When rationalising a binomial denominator (a + √b), multiply by its conjugate (a − √b).

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