Surds are numbers left in square root form that are used when detailed accuracy is required in a calculation. They are numbers which, when written in decimal form, would go on forever.
A fraction whose denominatorThe bottom part of a fraction. For ⅝, the denominator is 8, which represents 'eighths'. is a surd can be simplified by making the denominator rational. This process is called rationalising the denominator.
If the denominator has just one term that is a surd, the denominator can be rationalised by multiplying the numerator and denominator by that surd.
Example
Rationalise the denominator of \(\frac{\sqrt{8}}{\sqrt{6}}\).
The denominator can be rationalised by multiplying the numerator and denominator by √6.
\(\frac{5 \sqrt{3}}{6}\)Multiply the numerator and denominator by \(\sqrt{3}\). It is not wrong to multiply by \(2\sqrt{3}\) but you would need to simplify your answer at the end.
Rationalising the denominator when the denominator has a rational term and a surd
If the denominator of a fraction includes a rational number, add or subtract a surd, swap the + or – sign and multiply the numerator and denominator by this expression.
For example, if the denominator includes the bracket \((5 + 2\sqrt{3})\), then multiply the numerator and denominator by \((5 - 2\sqrt{3})\).
Example
Rationalise the denominator of \(\frac{5}{3-\sqrt{2}}\)
Rationalise the denominator by multiplying the numerator and denominator by \(3 + \sqrt{2}\).