Rationalising the denominator

Edexcel International GCSE Further Pure Mathematics· 1D· 12 min read

1. Rationalising Single Surd Denominators★★☆☆☆⏱ 4 min

✓ Calculator OK

When a denominator contains only a single surd of the form $a$ , you can rationalise it by multiplying both the numerator and denominator by $a$ . This works because $a \times a = a$ , a rational number, and you are effectively multiplying by 1 so the value of the expression does not change.

📘 Definition

Single surd denominator

A denominator consisting only of a radical (square root) term with no additional constants or added/subtracted terms

📐 Worked Example

Rationalise $\frac{10}{2}$ , leaving your answer in simplest form.

1
Multiply numerator and denominator by $2$ to eliminate the surd in the denominator
$\frac{10}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}}$
2
Simplify the denominator using $2 \times 2 = 2$
$\frac{10\sqrt{2}}{2}$
3
Divide numerator and denominator by their highest common factor of 2 to simplify fully
$5\sqrt{2}$

Exam tip:

Always simplify any resulting fraction after rationalising to avoid losing marks for incomplete working.

2. Rationalising Binomial Surd Denominators★★★☆☆⏱ 5 min

✓ Calculator OK

When a denominator is a binomial containing a surd of the form $a \pm b$ , you use the conjugate of the denominator to rationalise. Multiplying a binomial by its conjugate eliminates the surd term, as the cross terms cancel out when you expand the product using the difference of squares identity.

📘 Definition

Conjugate of a binomial surd

For a binomial of the form $a + b$ , the conjugate is $a - b$ ; for $a - b$ , the conjugate is $a + b$ . The product of a binomial and its conjugate is always rational.

📐 Worked Example

Rationalise $\frac{1}{2 - 3}$ , giving your answer in exact form.

1
Identify the conjugate of the denominator $2 - 3$ : this is $2 + 3$
2
Multiply numerator and denominator by the conjugate
$\frac{1}{2 - \sqrt{3}} \times \frac{2 + \sqrt{3}}{2 + \sqrt{3}}$
3
Expand the denominator using the difference of squares identity $(x - y) (x + y) = x^{2} - y^{2}$
$\frac{2 + \sqrt{3}}{2^2 - (\sqrt{3})^2} = \frac{2 + \sqrt{3}}{4 - 3} = 2 + \sqrt{3}$

Exam tip:

Use the difference of squares identity instead of FOIL to expand the denominator, to avoid common sign errors.

3. Combined Rationalisation Problems★★★★☆⏱ 3 min

✓ Calculator OK

Exam questions often require you to simplify surds first before rationalising, or combine rationalised terms with other expressions to get the final answer in the required form. Always check that your final answer has no common factors between numerator and denominator, and all surds are fully simplified.

📐 Worked Example

Simplify fully: $\frac{12}{8} + \frac{1}{3 + 2}$

1
Simplify $8 = 4 \times 2 = 22$ , then simplify the first term
$\frac{12}{2\sqrt{2}} = \frac{6}{\sqrt{2}}$
2
Rationalise the first term and simplify
$\frac{6}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{6\sqrt{2}}{2} = 3\sqrt{2}$
3
Rationalise the second term using its conjugate $3 - 2$
$\frac{1}{3 + \sqrt{2}} \times \frac{3 - \sqrt{2}}{3 - \sqrt{2}} = \frac{3 - \sqrt{2}}{9 - 2} = \frac{3 - \sqrt{2}}{7}$
4
Combine the terms over a common denominator and simplify
$3\sqrt{2} + \frac{3 - \sqrt{2}}{7} = \frac{21\sqrt{2} + 3 - \sqrt{2}}{7} = \frac{3 + 20\sqrt{2}}{7}$

✓ Quick check

What is the conjugate of $5 + 7$ ?
- $5 - 7$
- $- 5 + 7$
- $7 + 5$
- $7 - 5$
Rationalise $\frac{4}{3}$ : what is the simplified answer?
- $43$
- $\frac{4 3}{3}$
- $\frac{3}{4}$
- $123$
Reveal answer
1 —
Correct: multiply numerator and denominator by $3$ to get $\frac{4 3}{3}$ .

4. Common Pitfalls

Wrong move:

Only multiplying the denominator by $a$ or the conjugate, not the numerator

Why:

This changes the value of the entire expression, leading to an incorrect answer

Correct move:

Always multiply both numerator and denominator by the same term (equivalent to multiplying by 1) to preserve the expression's value

Wrong move:

Changing the sign of the constant term when finding the conjugate, e.g., using $- 2 + 3$ as the conjugate of $2 - 3$

Why:

The product will not be rational if you change the constant sign, so the surd in the denominator will not be eliminated

Correct move:

Only change the sign of the surd term when finding the conjugate: the conjugate of $2 - 3$ is $2 + 3$

Wrong move:

Failing to simplify the final fraction after rationalising, e.g., leaving $\frac{10 2}{2}$ as the final answer

Why:

Examiners require answers to be in simplest form, so you will lose marks for incomplete simplification

Correct move:

Always divide numerator and denominator by their highest common factor after rationalising to get the simplest form

Wrong move:

Expanding the denominator of a binomial surd using FOIL instead of the difference of squares formula

Why:

Manual expansion often leads to mistakes with negative signs when combining cross terms

Correct move:

Use the difference of squares identity $(x - y) (x + y) = x^{2} - y^{2}$ to expand the denominator quickly and accurately

5. Quick Reference Cheatsheet

Denominator Type	Rationalisation Method	Example
Single surd ( $\frac{1}{a}$ )	Multiply numerator & denominator by $a$	$\frac{1}{2} = \frac{2}{2}$
Binomial ( $\frac{1}{a + b}$ )	Multiply numerator & denominator by $a - b$	$\frac{1}{3 + 5} = \frac{3 - 5}{4}$
Binomial ( $\frac{1}{a - b}$ )	Multiply numerator & denominator by $a + b$	$\frac{1}{2 - 3} = 2 + 3$

6. Frequently Asked

Do I have to rationalise denominators in all 4PM1 exam answers?

Yes, unless explicitly stated otherwise. Examiners require final surd answers to have rational denominators to award full marks.

Can I use a calculator to verify my rationalisation work?

Yes, this topic is tested on calculator papers. You can compare the decimal value of the original expression and your rationalised result to confirm they match.

Going deeper

official_documentEdexcel 4PM1 SpecificationOfficial specification for Section 1 (Algebra)

What's Next

Now that you have mastered rationalising denominators, you can apply this skill to a wide range of topics in Edexcel IGCSE Further Pure Math, including coordinate geometry problems involving surd distances, trigonometry exact value calculations, and solving quadratic equations with surd roots. This skill is also a prerequisite for more advanced surd manipulation you will encounter if you progress to A-level Mathematics. Make sure you practice past paper questions to familiarise yourself with how rationalisation is tested in combination with other S1 topics.