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.
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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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