IGCSE Maths / Advanced algebra
Rationalising with conjugates
Edexcel International GCSE Maths (4MA1), chapter 23: Advanced algebra, specification point 1.4b. Free lesson.
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A conjugate is a bracket with the middle sign flipped. Multiplying the top and bottom of a fraction by the denominator’s conjugate makes the root terms cancel, leaving a rational denominator.
What a conjugate is
The conjugate of a bracket keeps the same two parts but changes the sign between them. This is useful when a denominator contains a bracket with a surd, because multiplying by its conjugate makes the surd disappear from the denominator. The conjugate of 3 + √5 is 3 − √5. The conjugate of 3 − √5 is 3 + √5.
Why the middle terms cancel
When you multiply conjugate brackets, the two middle terms have the same size but opposite signs. They cancel, while the two outside terms leave a whole number.
How to rationalise a denominator
To rationalise a denominator means to remove the root from the bottom of a fraction. Multiply the fraction by the denominator’s conjugate over itself. This fraction equals 1, so the value stays the same. Then simplify and cancel at the end.
Why the sign must change
Multiplying by the same bracket again does not rationalise the denominator. For example, squaring 3 + √5 gives 14 + 6√5, which still contains a root. The conjugate is needed because its opposite signs make the middle terms cancel. Use 3 − √5 with 3 + √5, not 3 + √5 again.
Check the result and finish simplifying
After rationalising, check that no root remains underneath. Also look for common factors to cancel. Leaving a common factor in the numerator and denominator means the answer is not fully simplified. For 4/(3 + √5), the denominator becomes 9 − 5 = 4. The factor of 4 cancels, leaving 3 − √5.