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Surds (Radicals): Simplifying and Rationalising

Simplify surds, calculate with them and rationalise denominators, with step-by-step examples. UK surds and US radicals, for GCSE Higher and Algebra 2.

  • ⏱ 3 min read
  • ✏️ 3 worked examples
  • 🎯 5 practice questions
  • 📶 Intermediate
GCSEA-LevelHigh School
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Curriculum fit: GCSE Maths Number (Higher tier). US: Algebra I and II radicals and rational exponents (Number and Quantity).

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What Is It?

A surd is a square root that cannot be written as a whole number or a simple fraction, such as √2 or √7. Surds are exact values: √2 is kept in this form instead of being rounded to 1.414.

In the US, these are called radicals. Surds follow a few rules that let you simplify them and combine them in calculations.

√a × √b = √(ab)√2 × √3 = √6√a ÷ √b = √(a/b)√6 ÷ √2 = √3√a × √a = a√5 × √5 = 51/√a = √a/a1/√2 = √2/2 (rationalised)
A surd is a root that cannot be written as an exact fraction. Rationalising means removing a surd from the bottom of a fraction.

Key Formulas and Ideas

  • √a × √b = √(ab)
  • √a ÷ √b = √(a/b)
  • √(a²) = a, so √a × √a = a
  • Rationalise a denominator: 1/√a = √a/a.

Worked Examples

Example 1

Simplify a Surd

Question

Simplify √50.

  1. Find the largest square factor: 50 = 25 × 2.
  2. √50 = √25 × √2 = 5√2.

✓ Answer 5√2

Example 2

Rationalise

Question

Rationalise 6/√2.

  1. Multiply top and bottom by √2.
  2. 6√2 / (√2 × √2) = 6√2/2.
  3. Simplify: 3√2.

✓ Answer 3√2

Example 3

Expanding Brackets

Question

Expand and simplify (2 + √3)(2 − √3).

  1. Multiply: 4 − 2√3 + 2√3 − 3.
  2. The √3 terms cancel: 4 − 3 = 1.

✓ Answer 1

Common Mistakes to Avoid

  • Writing √(a + b) as √a + √b. It is not equal.
  • Leaving a square factor inside the surd, such as √18 instead of 3√2.
  • Rounding a surd when the question asks for an exact answer.
  • Forgetting to rationalise when asked.

Practice Questions

0 of 5 answered
Q1Simplify √18.
Q2Work out √5 × √5.
Q3Simplify 6/√2.
Q4Expand (1 + √2)².
Q5Simplify √12 + √3.

Frequently Asked Questions

Is √9 a surd?

No. √9 = 3, a whole number, so it is not a surd.

Why leave answers as surds?

Surds are exact. Rounding introduces small errors that can grow in later steps.

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Sources and Review

Reviewed by StudyMorrow editorial review on 5 Nov 2026.