Rationalising Denominators (SQA National 5 Maths): Revision Note

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. $\frac{4}{\sqrt{5}}$ or $\sqrt{\frac{2}{3}}=\frac{\sqrt{2}}{\sqrt{3}}$

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

  • E.g. $\frac{4\sqrt{5}}{5}$ or $\frac{\sqrt{6}}{3}$

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

How do I rationalise a denominator?

• If the denominator is a surd:

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

    • $\frac{a}{\sqrt{b}}= \frac{a}{\sqrt{b}} \times \frac{\sqrt{b}}{\sqrt{b}}$

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

    • $\sqrt{b}\times \sqrt{b}=b$ so the denominator is no longer a surd

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

    • $\frac{a\sqrt{b}}{b}$