If you're studying for the Higher Tier GCSE Maths exam (with AQA, Edexcel, or OCR), you've likely encountered surds. These are irrational numbers like √2, √5, and √11 that we leave in their exact form. A key skill for working with surds is rationalising the denominator. This post will explain what it is, why we do it, and give you a clear, step-by-step method to master it.
What Does 'Rationalise the Denominator' Mean?
The denominator is the bottom part of a fraction. To 'rationalise' it means to eliminate any surds from the denominator, leaving it as a rational number (like 2, 5, or 13). We do this because it's considered mathematically neater and is often required in exam answers.
For example, 1√2 has a surd in the denominator. We rationalise it to get √22.
Why Do We Rationalise the Denominator?
Standard Form: It's the conventional, accepted way to present your final answer in GCSE Maths.
Easier Calculations: It can make adding, subtracting, or comparing fractions with surds much simpler.
- Exam Marks: You will often be explicitly instructed to "simplify your answer" or "rationalise the denominator," and marks depend on it!
The Golden Rule
To rationalise the denominator, we multiply the numerator and the denominator of the fraction by the same surd (or expression). This is equivalent to multiplying by 1, so the value of the fraction doesn't change.
Type 1: A Single Surd in the Denominator
This is the simplest case. The denominator is just one surd, like √a.
Step-by-Step Method:
Identify the surd in the denominator (e.g., √5).
Multiply both the top and bottom of the fraction by that same surd.
- Simplify the new fraction.
Example 1: Rationalise 3√5
3√5 × √5/√5 = 3√5/√5 × √5 = 3√55
Why does this work? Because √5 × √5 = 5 (the square root undoes the square).
Example 2: Rationalise 7√2 / √3
7√2/√3 × √3/√3 = 7√2√33 = 7√63
(Remember: √2 × √3 = √6).
Type 2: A Binomial Denominator (Two Terms, Including a Surd)
This is a more advanced GCSE question. The denominator has two terms, like (a + √b) or (√a - b).
Key Concept: We use the difference of two squares. Remember: (x + y)(x - y) = x² - y². This eliminates the surd terms.
Step-by-Step Method:
Identify the denominator (e.g., 2 + √3).
Find its conjugate. This is the same expression but with the opposite sign between the terms (e.g., 2 - √3).
Multiply both the top and bottom of the fraction by this conjugate.
Expand the brackets carefully (using FOIL/expanding double brackets) on the top and bottom.
- Simplify the resulting fraction fully.
flowchart TD
A["Start: Fraction with binomial denominator e.g. 5/(2+sqrt(3))"] --> B["Find the conjugate of the denominator. Change the middle sign: 2-sqrt(3)"]
B --> C["Multiply numerator and denominator by the conjugate"]
C --> D["Expand brackets in numerator and denominator"]
D --> E["Simplify denominator using a^2 - b^2. Surd cancels!"]
E --> F["Simplify numerator and reduce fraction if possible"]
F --> G["End: Rationalised fraction"]
Example 3: Rationalise 5 / (2 + √3)
Denominator: (2 + √3). Its conjugate is (2 - √3).
- Multiply top and bottom:
5(2+√3) × 2-√32-√3
Numerator: 5 × (2 - √3) = 10 - 5√3.
Denominator: (2 + √3)(2 - √3) = (2)² - (√3)² = 4 - 3 = 1. (The surd vanishes!)
- Result:
10 - 5√31 = 10 - 5√3.
Example 4: Rationalise (√6) / (√5 - 2)
Denominator: (√5 - 2). Its conjugate is (√5 + 2).
- Multiply:
√6/(√5-2) × √5+2√5+2
Numerator: √6 × (√5 + 2) = √30 + 2√6.
Denominator: (√5 - 2)(√5 + 2) = (√5)² - (2)² = 5 - 4 = 1.
- Result:
√30 + 2√6.
Common Mistakes to Avoid
Forgetting to multiply the numerator: You must multiply both parts of the fraction.
Incorrect conjugate: For (a + √b), the conjugate is (a - √b), not (-a - √b).
Expanding errors: Be meticulous when expanding brackets, especially with negatives.
- Not fully simplifying: Always check if the final answer can be simplified (e.g., cancelling common factors).
Practice Questions
Try these, then check your answers below.
Rationalise 8√2
Rationalise 3√5√10
Rationalise 4 / (1 - √7)
- Rationalise
(√3 + 1) / (√3 - 1)
Answers:
8√22 = 4√2
3√5√1010 = 3√5010 = 3×5√210 = 15√210 = 3√22
[4(1+√7)] / [(1-√7)(1+√7)] = 4+4√71-7 = 4+4√7-6 = -2+2√73
[(√3+1)(√3+1)] / [(√3-1)(√3+1)] = 3+2√3+13-1 = 4+2√32 = 2+√3
Summary
Rationalising the denominator is a core GCSE skill. Remember:
Single surd? Multiply top and bottom by that surd.
Two terms (binomial)? Multiply top and bottom by the conjugate.
- Always simplify your final answer.
- With practice, this process will become second nature. Good luck with your revision!