Simplifying Fractions with Surds (WJEC GCSE Maths & Numeracy (Double Award): Higher): Revision Note

Simplifying Fractions with Surds

How do I simplify fractions containing surds?

• If the denominator is a surd:

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

    • $\frac{a\sqrt{b}}{b}$

• An alternate method is to rewrite the numerator in terms of the denominator, then cancel common factors

  • E.g. the numerator of $\frac{6}{\sqrt{2}}$ can be rewritten in terms of $\sqrt{2}$

    • $\frac{6}{\sqrt{2}}=\frac{3\times 2}{\sqrt{2}}=\frac{3\sqrt{2}\sqrt{2}}{\sqrt{2}}$

    • because $2=\sqrt{2}\times \sqrt{2}$

  • Then cancel out the common factors on the top and bottom

    • $\frac{6}{\sqrt{2}}=\frac{3\sqrt{2}\cancel{\sqrt{2}}}{\cancel{\sqrt{2}}}=3\sqrt{2}$

  • This method will not work for all cases

    • E.g. to simplify $\frac{3}{\sqrt{2}}$ you would need to multiply by $\frac{\sqrt{2}}{\sqrt{2}}$

• You may have to simplify the numerator first, often by expanding

  • See the worked example below