Rationalising the Denominator

A fraction is tidier — and easier to compare or add — when there is no surd on the bottom. To rationalise the denominator, multiply the top and bottom by the surd in the denominator. Since you multiply by the same thing top and bottom, you are really multiplying by 1, so the value of the fraction is unchanged:

\frac{1}{\sqrt{2}} = \frac{1}{\sqrt{2}}\times\frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}

This works because \sqrt{2}\times\sqrt{2} = 2: the root on the bottom is gone, replaced by a whole number.

To clear a surd from the bottom of a fraction:

multiply the numerator and denominator by the denominator's surd;
\sqrt{a}\times\sqrt{a} = a, so the root disappears from the bottom;
the fraction's value is unchanged, because you multiplied by 1;
for a denominator like (a + \sqrt{b}), multiply top and bottom by its conjugate (a - \sqrt{b}) instead.

For example:

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

and using a conjugate:

\frac{1}{2 + \sqrt{3}} = \frac{1}{2 + \sqrt{3}}\times\frac{2 - \sqrt{3}}{2 - \sqrt{3}} = \frac{2 - \sqrt{3}}{4 - 3} = 2 - \sqrt{3}