Rationalising Denominators (Cambridge (CIE) IGCSE International Maths)

Rationalising Denominators

What does rationalising the denominator mean?

• If a fraction has a denominator containing a surd then it has an irrational denominator

  • E.g. or

• The fraction can be rewritten as an equivalent fraction, but with a rational denominator

  • E.g. or

• The numerator may contain a surd, but the denominator is rationalised

How do I rationalise simple denominators?

• If the denominator is a surd:

  • Multiply the top and bottom of the fraction by the surd on the denominator

    • 

    • This is equivalent to multiplying by 1, so does not change the value of the fraction

    • so the denominator is no longer a surd

  • Multiply the fractions as you would usually, and simplify if needed

    • 

How do I rationalise harder denominators?

• If the denominator is an expression containing a surd:

  • For example  

  • Multiply the top and bottom of the fraction by the expression on the denominator, but with the sign changed

    • 

    • This is equivalent to multiplying by 1, so does not change the value of the fraction

  • Multiply the fractions as you would usually (use brackets to help)

    • 

  • Expand any brackets, and simplify

    • 

  • You can use the difference of two squares to expand the denominator quickly

    • 

    • This is what makes the denominator rational

  • Simplify

    •