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AQASpec N8Higher tier~25 min

Surds

Mathematics · Topic revision workspace

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Topic overview

What you need to know

Surds is a key part of GCSE Mathematics. Build fluency with the method, then apply it to unfamiliar and multi-step problems. An irrational root that cannot be simplified to a whole number, e.g. √2, √3, √5. Express the number under the root as a product involving a perfect square. E.g. √12 = √(4×3) = 2√3. √a × √b = √(ab). Multiply the numbers under the roots. E.g. √3 × √5 = √15.

Exam tip

Write every stage of your method. Method marks can still be earned when the final answer is wrong.

Revision notes

Core ideas

Secure these ideas first. Say each definition in your own words, then connect it to the topic overview.

  • Surd: An irrational root that cannot be simplified to a whole number, e.g. √2, √3, √5.
  • Simplifying Surds: Express the number under the root as a product involving a perfect square. E.g. √12 = √(4×3) = 2√3.
  • Multiplying Surds: √a × √b = √(ab). Multiply the numbers under the roots. E.g. √3 × √5 = √15.
  • Dividing Surds: √a ÷ √b = √(a/b). Divide the numbers under the roots.

Revision notes

Apply it in the exam

The exam will rarely ask for an isolated definition. Practise selecting the right idea and using it as part of a complete explanation or method.

  • Adding/Subtracting Surds: You can only add or subtract surds with the same root. E.g. 3√2 + 5√2 = 8√2.
  • Rationalising the Denominator: Remove the surd from the denominator by multiplying top and bottom by the surd. E.g. 1/√3 = √3/3.
  • Rationalising (Two Terms): For denominators like (a + √b), multiply by the conjugate (a − √b) to eliminate the surd.
  • Expanding Double Brackets with Surds: Use FOIL to expand. E.g. (2+√3)(1+√3) = 2 + 2√3 + √3 + 3 = 5 + 3√3.

Core knowledge

Key facts for Surds

1

Surd

An irrational root that cannot be simplified to a whole number, e.g. √2, √3, √5.

2

Simplifying Surds

Express the number under the root as a product involving a perfect square. E.g. √12 = √(4×3) = 2√3.

3

Multiplying Surds

√a × √b = √(ab). Multiply the numbers under the roots. E.g. √3 × √5 = √15.

4

Dividing Surds

√a ÷ √b = √(a/b). Divide the numbers under the roots.

5

Adding/Subtracting Surds

You can only add or subtract surds with the same root. E.g. 3√2 + 5√2 = 8√2.

6

Rationalising the Denominator

Remove the surd from the denominator by multiplying top and bottom by the surd. E.g. 1/√3 = √3/3.

7

Rationalising (Two Terms)

For denominators like (a + √b), multiply by the conjugate (a − √b) to eliminate the surd.

8

Expanding Double Brackets with Surds

Use FOIL to expand. E.g. (2+√3)(1+√3) = 2 + 2√3 + √3 + 3 = 5 + 3√3.

Active recall

Close the notes and answer these

  1. 1.Without looking, explain surd and give one example or consequence.
  2. 2.Without looking, explain simplifying surds and give one example or consequence.
  3. 3.Without looking, explain multiplying surds and give one example or consequence.
  4. 4.Without looking, explain dividing surds and give one example or consequence.

Content reviewed 23 July 2026 against the current linked specification.

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