Rationalising denominators

A fraction whose 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.

\(\frac{\sqrt{8} \times \sqrt{6}}{\sqrt{6} \times \sqrt{6}} = \frac{\sqrt{48}}{6} = \frac{\sqrt{(16 \times 3)}}{6} = \frac{4 \sqrt{3}}{6} = \frac{2 \sqrt{3}}{3}\)

Question

Rationalise the denominator of the following:

\(\frac{1}{\sqrt{2}}\)
\(\frac{\sqrt{3}}{\sqrt{2}}\)
\(\frac{5}{2 \sqrt{3}}\)

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 \(3 + \sqrt{2}\).

\(\frac{5}{3-\sqrt{2}} = \frac{5(3+\sqrt{2})}{(3-\sqrt{2})(3+\sqrt{2})} = \frac{15+5\sqrt{2}}{9+3\sqrt{2}-3\sqrt{2}-2} = \frac{15+5\sqrt{2}}{7}\)

Question

Rationalise the denominator of \(\frac{11}{6-2\sqrt{5}}\) giving your answer in its simplest form.

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Surds - Higher - OCRRationalising denominators