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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.

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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

  1. Find the largest perfect square factor of 72: 36 × 2 = 72
  2. √72 = √(36 × 2) = √36 × √2
  3. = 6√2
Answer6√2
Example 2

Rationalise the denominator of 5/√3

Working

  1. Multiply numerator and denominator by √3
  2. 5/√3 × √3/√3 = 5√3 / (√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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