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Rationalising denominators with surds

We don't like having surds in the denominator of fractions. Removing the surd from the denominator is called rationalisation of the denominator.

We prefer $$\displaystyle\frac{\sqrt{2}}{2}$$ to $$\displaystyle\frac{1}{\sqrt{2}}$$ and $$\displaystyle\frac{\sqrt{6}}{3}$$ to $$\displaystyle\frac{\sqrt{2}}{\sqrt{3}}$$.

  • When the denominator is a simple surd, we multiply both the denominator and the numerator by the surd $$$\frac{2}{\Tblue{\sqrt{10}}} = \frac{2\Tgreen{\sqrt{10}}}{\Tblue{\sqrt{10}}\times \Tgreen{\sqrt{10}}} = \frac{2\sqrt{10}}{10} = \frac{\sqrt{10}}{5}.$$$
  • When the denominator is the sum of two surds, we multiply by the conjugate surd

    The conjugate of $$\Tblue{\sqrt{3} + \sqrt{2}}$$ is $$\Tgreen{\sqrt{3}-\sqrt{2}}$$. This helps us to simplify the following fraction. $$$\frac{1}{\Tblue{\sqrt{3} + \sqrt{2}}} = \frac{1}{\Tblue{\sqrt{3} + \sqrt{2}}} \times \frac{\Tgreen{\sqrt{3} - \sqrt{2}}}{\Tgreen{\sqrt{3} - \sqrt{2}}} = \sqrt{3} - \sqrt{2}$$$