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Rationalised Denominator - GCSE Maths Definition

Reviewed by: Roger B

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What is a rationalised denominator?

In fractions like fraction numerator 12 over denominator square root of 3 end fraction or fraction numerator 1 over denominator 7 minus square root of 5 end fraction, the denominators are not rational numbers. That is because the denominators contain surds, which are irrational numbers.

When a number like that is rewritten without any surds in the denominator (or with no denominator at all), we say that the denominator has been rationalised.

For example, fraction numerator 12 over denominator square root of 3 end fraction equals 4 square root of 3, or fraction numerator 1 over denominator 7 minus square root of 5 end fraction equals fraction numerator 7 plus square root of 5 over denominator 44 end fraction

How do I rationalise a denominator?

If the denominator of a fraction is a surd on its own, then multiply the fraction top and bottom by that surd. For example,

fraction numerator 12 over denominator square root of 3 end fraction equals fraction numerator 12 over denominator square root of 3 end fraction cross times fraction numerator square root of 3 over denominator square root of 3 end fraction equals fraction numerator 12 square root of 3 over denominator 3 end fraction equals 4 square root of 3

If the denominator is an expression containing a surd, then multiply the fraction top and bottom by the expression in the denominator, but with the sign changed. For example,

table row cell fraction numerator 1 over denominator 7 minus square root of 5 end fraction end cell equals cell fraction numerator 1 over denominator 7 minus square root of 5 end fraction cross times fraction numerator 7 plus square root of 5 over denominator 7 plus square root of 5 end fraction end cell row blank equals cell fraction numerator 7 plus square root of 5 over denominator open parentheses 7 minus square root of 5 close parentheses open parentheses 7 plus square root of 5 close parentheses end fraction end cell row blank equals cell fraction numerator 7 plus square root of 5 over denominator 49 plus 7 square root of 5 minus 7 square root of 5 minus 5 end fraction end cell row blank equals cell fraction numerator 7 plus square root of 5 over denominator 44 end fraction end cell end table

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Roger B

Reviewer: Roger B

Expertise: Maths Content Creator

Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.

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