Rationalising the denominator means rewriting a fraction so that no surd appears on the bottom line. Mathematicians prefer rational denominators because they make arithmetic cleaner and comparison between fractions easier. The method uses multiplication by a carefully chosen form of 1 — either the surd itself or its conjugate — which leaves the value unchanged while removing the surd below.
Why do we rationalise the denominator?
A surd in the denominator makes fractions harder to add and compare. For example, 1/√2 looks different from 3/√2 even though their sum 4/√2 = 4√2/2 = 2√2 is easy once the denominators are rational. GCSE mark schemes also typically expect answers with rational denominators unless stated otherwise, so failing to rationalise loses marks even if the value is correct.
The key principle: multiplying the numerator and denominator of a fraction by the same non-zero expression gives an equivalent fraction — just as multiplying 1/2 by 3/3 gives 3/6. Choosing what to multiply by is the skill.
How do you rationalise a simple surd denominator?
When the denominator is a single surd √a, multiply numerator and denominator by √a.
Worked example: Rationalise 5/√3.
- Multiply top and bottom by √3: (5 × √3) / (√3 × √3).
- Simplify: √3 × √3 = 3.
- Result: 5√3 / 3.
The denominator is now the rational number 3.
Second example: Rationalise 6/√8.
- Multiply by √8/√8: (6√8) / (√8 × √8) = 6√8 / 8.
- Simplify 6/8 = 3/4 and √8 = 2√2: result = 3 × 2√2 / 4 = 3√2 / 2.
It is good practice to simplify both the surd and the coefficient at the end.
What is a conjugate pair and why does it help?
When the denominator contains a binomial surd of the form a + √b or a − √b, multiplying by √b alone does not help — it introduces new surds in the denominator. Instead, multiply by the conjugate: swap the sign between the two terms.
- The conjugate of (3 + √5) is (3 − √5).
- The conjugate of (2 − √7) is (2 + √7).
The key result is: (a + √b)(a − √b) = a² − b, which is always rational. This is the difference of two squares identity, used here to eliminate the surd entirely.
How do you rationalise a binomial surd denominator?
Worked example: Rationalise 2 / (3 + √5).
- The conjugate of (3 + √5) is (3 − √5). Multiply numerator and denominator by (3 − √5): 2(3 − √5) / ((3 + √5)(3 − √5))
- Expand the denominator: (3)² − (√5)² = 9 − 5 = 4.
- Expand the numerator: 2(3 − √5) = 6 − 2√5.
- Result: (6 − 2√5) / 4 = (3 − √5) / 2.
Second example: Rationalise (1 + √3) / (√3 − 1).
- Conjugate of (√3 − 1) is (√3 + 1). Multiply top and bottom by (√3 + 1): (1 + √3)(√3 + 1) / ((√3 − 1)(√3 + 1))
- Denominator: (√3)² − 1² = 3 − 1 = 2.
- Numerator — expand using FOIL: 1×√3 + 1×1 + √3×√3 + √3×1 = √3 + 1 + 3 + √3 = 4 + 2√3.
- Result: (4 + 2√3) / 2 = 2 + √3.
Summary of the two methods
| Denominator type | Multiply by | Result in denominator |
|---|---|---|
| √a (single surd) | √a | a (rational) |
| a + √b (binomial surd) | a − √b (conjugate) | a² − b (rational) |
| a − √b (binomial surd) | a + √b (conjugate) | a² − b (rational) |
What are the most common errors?
- Multiplying only the denominator, not the numerator. Both must be multiplied by the same expression, otherwise the fraction's value changes.
- Expanding the conjugate product incorrectly. (a + √b)(a − √b) = a² − b, not a² − 2a√b + b. The middle terms cancel — that is the whole point. Avoid treating it as an ordinary bracket expansion.
- Forgetting to simplify the final fraction. After rationalising, look for common factors between numerator and denominator. In the example above, (6 − 2√5)/4 simplifies to (3 − √5)/2 by dividing by 2.
Frequently asked questions
How does rationalising the denominator relate to the difference of two squares?
The identity (a + b)(a − b) = a² − b² is the algebraic engine behind conjugate rationalisation. When b = √c, the identity becomes (a + √c)(a − √c) = a² − c, which is always rational. Recognising the conjugate pair as a difference-of-two-squares problem is the conceptual link that makes the method easy to recall.
Does rationalising change the value of the fraction?
No. Multiplying numerator and denominator by the same non-zero quantity is equivalent to multiplying by 1. The fraction is rewritten in an equivalent form — the same rational point on the number line, just expressed differently. You can always check by converting both forms to decimals: 5/√3 ≈ 2.887 and 5√3/3 = 5 × 1.732/3 ≈ 2.887. ✓
When would GCSE questions ask you to rationalise a denominator?
Rationalising appears in questions on surds (GCSE Higher), algebraic fractions, and sometimes trigonometry when exact values produce surd denominators (e.g. tan 60° = √3, so 1/tan 60° = 1/√3 = √3/3). The instruction is usually "simplify" or "write in the form a + b√c".
Can the numerator contain surds after rationalising?
Yes — in fact, the purpose of rationalising is to move the surd from the denominator to the numerator, where it is easier to work with. After rationalising 5/√3, the surd √3 appears in the numerator: 5√3/3. This is the intended outcome, not a problem to fix.
For Socratic surd practice at GCSE Higher with Professor Pi, visit aitutors.me.