GCSE MATHS • HIGHER TIER

Rationalising the Denominator Made Simple

Remove surds from denominators and simplify exact values confidently.

Start Learning
Practice Questions
Download Worksheet

Clear method

Exam practice

Worked proofs

Rationalising the Denominator in Four Moves

1

Identify

Find the surd in the denominator.

2

Multiply

Multiply the numerator and denominator by the same surd.

3

Simplify

Use surd multiplication to simplify the denominator and numerator.

4

Check

Make sure no surd remains in the denominator.

What You’ll Learn

Understand what rationalising the denominator means.

Rationalise fractions with a single surd in the denominator.

Use the rule √a × √a = a.

Simplify surds before rationalising when useful.

Work with surds in both the numerator and denominator.

Recognise and avoid common GCSE mistakes.

Key Rules

When a denominator contains a single surd, multiply both the numerator and denominator by that surd. This creates a square number in the denominator, which can then be simplified.

√a × √a = a

Worked Proof: Rationalising the Denominator

Step-by-Step Method

  • 1. Identify the surd in the denominator.
  • 2. Multiply the numerator and denominator by the same surd.
  • 3. Simplify the denominator.
  • 4. Simplify the numerator if possible.
  • 5. Check that no surds remain in the denominator.

More Worked Examples

1

Simplify 1/√2.

Multiply top and bottom by √2.

(1 × √2)/(√2 × √2)
= √2/2
Answer : √2/2

2

Simplify 3/√5.

Multiply by √5.

(3√5)/(√5 × √5)
= 3√5/5
Answer : 3√5/5

3

Simplify 2/√3.

Multiply by √3.

(2√3)/(√3 × √3)
= 2√3/3
Answer : 2√3/3

4

Simplify 5/√7.

Multiply by √7.

(5√7)/(√7 × √7)
= 5√7/7
Answer : 5√7/7
  • Common GCSE Mistakes
  • Multiplying only the denominator instead of both parts.
  • Forgetting to simplify the final answer.
  • Leaving a surd in the denominator.
  • Skipping useful surd simplification before rationalising.
  • Exam Tips
  • Rationalise whenever a surd remains in the denominator.
  • Multiply top and bottom by the same surd.
  • Use √a × √a = a carefully.
  • Simplify surds before rationalising when possible.
  • Check the final denominator contains no surd.
  • Leave exact answers in simplified surd form.

Skills Used in Rationalising the Denominator

Surds

Exact Values

Algebraic Manipulation

Fractions

Square Roots

Multiplication

Ready to Practise?

Printable Worksheet

Download and print practice questions with answers.

Open Resource

Interactive Quiz

Test your knowledge with instant feedback and hints.

Open Resource

Exam-Style Questions

Timed questions to build confidence for the real exam.

Open Resource

Watch the Rationalising the Denominator Tutorial

Step-by-step walkthroughs of Rationalising the Denominator with clear methods and exam tips.

Watch Video

Frequently Asked Questions

It means rewriting a fraction so that no surd remains in the denominator.

Because √a × √a = a, which turns the denominator into a rational number.

Yes. Multiplying both by the same value keeps the fraction equivalent.

Yes, when possible. The source specifically recommends simplifying surds first if it makes the expression easier.

Yes. Rationalising only requires removing surds from the denominator.

It builds on surds, exact values and algebraic manipulation, which are assessed at Higher Tier.