Rationalising the Denominator –

Rationalising the denominator is a Higher tier skill that appears regularly on GCSE Maths exams. Examiners expect answers to have no surds in the denominator, so knowing how to remove them is essential for earning full marks on surd questions.

What Is Rationalising the Denominator?

Rationalising the denominator means rewriting a fraction so that the denominator contains no surds — only rational numbers. The value of the fraction does not change; you are simply expressing it in a standard form.

There are two cases. When the denominator is a single surd like √3, you multiply top and bottom by that surd. When the denominator is a two-term expression like (3 + √2), you multiply top and bottom by the conjugate — the same expression with the opposite sign on the surd, so (3 − √2).

The conjugate method works because of the difference of two squares: (a + √b)(a − √b) = a² − b, which eliminates the surd from the denominator.

Key Formulas

Step-by-Step Method

1. Single surd denominator: Multiply the numerator and denominator by the surd in the denominator.
2. Two-term denominator (e.g. a + √b): Multiply numerator and denominator by the conjugate (a − √b).
3. Expand the denominator — the surd terms will cancel, leaving a rational number.
4. Expand the numerator and simplify.
5. Cancel any common factors between numerator and denominator.

Worked Example 1 — Foundation Level

Question: Rationalise the denominator of 6/√3. Give your answer in its simplest form.

Working:

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

Step 2 — Simplify: 6√3 / 3 = 2√3.

Answer: 2√3

Worked Example 2 — Higher Level

Question: Rationalise the denominator of 5/(2 + √3). Give your answer in the form a + b√3.

Working:

Step 1 — The conjugate of (2 + √3) is (2 − √3). Multiply top and bottom by (2 − √3).

Step 2 — Denominator: (2 + √3)(2 − √3) = 4 − 3 = 1.

Step 3 — Numerator: 5(2 − √3) = 10 − 5√3.

Step 4 — The fraction becomes (10 − 5√3) / 1 = 10 − 5√3.

Answer: 10 − 5√3

Worked Example 3 — Exam Style

Question: Show that 12/(3 − √5) can be written as a(3 + √5) where a is an integer to be found. (3 marks)

Working:

Step 1 — Multiply top and bottom by the conjugate (3 + √5).

Step 2 — Denominator: (3 − √5)(3 + √5) = 9 − 5 = 4.

Step 3 — Numerator: 12(3 + √5).

Step 4 — The fraction becomes 12(3 + √5) / 4 = 3(3 + √5).

So a = 3.

Answer: a = 3

Common Mistakes

• Multiplying only the numerator by the surd. You must multiply both the numerator and the denominator to keep the fraction equivalent. Multiplying top and bottom by the same thing is equivalent to multiplying by 1.
• Using the wrong conjugate. The conjugate of (a + √b) is (a − √b), not (a + √b). Using the same expression will not eliminate the surd.
• Forgetting to simplify after rationalising. After clearing the surd from the denominator, check whether the numerator and denominator share common factors.

Exam Tips

• Questions often combine simplifying surds with rationalising. Simplify the surds first, then rationalise.
• When the denominator multiplies out to 1 (as in the worked example above), the answer simplifies very neatly. Watch for this — it can save time.
• Write the conjugate multiplication step clearly. This earns a method mark even if you make an arithmetic error afterwards.

Practice Questions

Q1 (Higher): Rationalise the denominator of 10/√5. Simplify your answer.

Q2 (Higher): Rationalise the denominator of 4/(1 + √3). Give your answer in the form a + b√3.

Q3 (Higher): Rationalise the denominator of (3 + √2)/(√2). Simplify fully.

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Related Topics

• Surds
• Simplifying Surds
• Indices and Index Laws

Summary

• Rationalising the denominator removes surds from the bottom of a fraction.
• For a single surd denominator, multiply top and bottom by that surd.
• For a two-term denominator, multiply by the conjugate to use the difference of two squares.
• Always simplify your final answer by cancelling common factors.
• Show the conjugate multiplication step clearly to earn method marks in exams.