Rationalising the denominator simplifies expressions involving surds, ensuring the denominator of a fraction does not contain a square root. This technique is essential for IGCSE exams, facilitating easier manipulation and standardisation of surd expressions.

Introduction to Rationalising the Denominator

The process involves multiplying the numerator and denominator by a suitable surd to eliminate the square root from the denominator. It's a key skill for simplifying expressions and preparing for exams.

Techniques for Rationalising the Denominator

Single Surd in the Denominator

To remove a single surd, multiply the numerator and denominator by that surd.

Example 1: Rationalise $\frac{1}{\sqrt{3}}$.

Solution:

Multiply by $\frac{\sqrt{3}}{\sqrt{3}}$.

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Denominator with Two Terms (Conjugate Pair)

For denominators like $a + \sqrt{b}$ or $a - \sqrt{b}$, use the conjugate to rationalise.

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Example 2: Rationalise $\frac{4}{1+\sqrt{5}}$.

Solution:

1. Conjugate is $1-\sqrt{5}$.

2. Multiply by $\frac{1-\sqrt{5}}{1-\sqrt{5}}$.

Practice Questions

Question 1: Rationalise $\frac{3}{2\sqrt{7}}$.

Solution:

Multiply by $\frac{\sqrt{7}}{\sqrt{7}}$.

Question 2: Rationalise $\frac{5}{3-\sqrt{2}}$.

Solution:

1. Conjugate is $3+\sqrt{2}$.

2. Multiply by $\frac{3+\sqrt{2}}{3+\sqrt{2}}$.

Key Points to Remember

• Use the conjugate for denominators with two terms.
• Maintain the fraction's value by multiplying the numerator and denominator by the same expression.
• Always simplify your final answer for clarity and conciseness.