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 鈪, the denominator is 8, which represents 'eighths'. is a surd can be simplified by making the denominator rational numberA number that can be written in fraction form. This includes integers, terminating decimals, repeating decimals and fractions.. 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 numeratorThe top part of a fraction. For 鈪 , the numerator is 5. 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 鈭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 鈥 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 \(3 + \sqrt{2}\).