IGCSE Maths / Advanced algebra
Rationalising a denominator
Edexcel International GCSE Maths (4MA1), chapter 23: Advanced algebra, specification point 1.4b. Free lesson.
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Rationalising a denominator means rewriting a fraction so there is no root on the bottom, without changing its value. The final answer must also be fully simplified.
What rationalising a denominator means
The denominator is the bottom of a fraction. A surd is a root that is not a whole number. Rationalising means moving the root off the denominator, because a root left there is not accepted as a final answer. The fraction keeps exactly the same value. Only the way it is written changes. Rationalising a denominator is part of manipulating surds.
How to remove the root from the bottom
Multiply the top and bottom by the root on the bottom. This is allowed because the root divided by itself equals 1, so the fraction's value stays the same. The denominator becomes a whole number because a root multiplied by itself gives the number inside it. For example, √3 ÷ √3 = 1, so multiplying by √3/√3 does not change the value. Once the root has moved, simplify the fraction by cancelling any common factors in the plain numbers.
Rationalising 6/√3
Multiply top and bottom by √3. The denominator becomes 3, and then the plain numbers can be cancelled. That last simplification is part of the final answer.
When the answer is still a fraction
Sometimes the plain numbers cannot be cancelled completely. That is fine, as long as the denominator has no root and the fraction is simplified as far as possible. Keep the root with the numerator and cancel only the plain numbers.
Finish by simplifying
Getting the root off the bottom is not the whole job. Check whether the plain numbers on the top and bottom share a factor. Leaving a fraction such as 6√3/3 unfinished can lose a mark. For 12/√6, multiplying top and bottom by √6 gives 12√6/6. Cancel 12 and 6 to get 2√6.