Rationalising denominators - Higher

A fraction with a surd in the denominator can be simplified so that the denominator is an integer. This is called rationalising the denominator.

Example

Simplify \frac{\sqrt{8}}{\sqrt{6}}

First simplify \sqrt{8}.

\sqrt{8} = \sqrt{4 \times 2}

= \sqrt{4} \times \sqrt{2}

= 2\sqrt{2}

Then multiply the numerator and denominator by \sqrt{6}.

\frac{2\sqrt{2} \times \sqrt{6}}{\sqrt{6} \times \sqrt{6}}

= \frac{2\sqrt{12}}{6}

= \frac{4\sqrt{3}}{6} = \frac{2\sqrt{3}}{3}

Question

Rationalise the denominator of the following:

  1. \frac{1}{\sqrt{2}}
  2. \frac{\sqrt{3}}{\sqrt{2}}
  3. \frac{5}{2 \sqrt{3}}
  1. \frac{\sqrt{2}}{2} (Multiply by a fraction, the equivalent of 1 that is in the surd form of the denominator.)
  2. \frac{\sqrt{6}}{2}
  3. \frac{5 \sqrt{3}}{6}.
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