Just been trying to think of an example of rationalising and the only thing I could think of is rationalising the denominator or a fraction containing a surd.

As you may or may not know, a surd is a number such as $\displaystyle \sqrt{2}$ that cannot be express as a 'normal' number, ie. you can't write it as a finite decimal.

Fractions containing surds as a denominator aren't considered to be 'elegent' as my maths tutor once put it. For example:

$\displaystyle \frac{3}{\sqrt{2}}$ can be rationalised to become $\displaystyle \frac{3\sqrt{2}}{2}$ by multiplying by $\displaystyle \sqrt{2}$, this is considered rationalised.

Another example is where you have a fraction such as $\displaystyle \frac{3}{\sqrt{2} + \sqrt{5}}$.

To deal with this you use the

Difference of two squares identity, multiplying top and bottom by $\displaystyle (\sqrt{2} - \sqrt{5})$.

This gives you $\displaystyle \frac{3(\sqrt{2} - \sqrt{5})}{(\sqrt{2} + \sqrt{5})(\sqrt{2} - \sqrt{5})}$

$\displaystyle \frac{3(\sqrt{2} - \sqrt{5})}{\sqrt{2}^2 + \sqrt{10} -\sqrt{10} - \sqrt{5}^2}$

$\displaystyle \frac{3(\sqrt{2} - \sqrt{5})}{-3}$

$\displaystyle \sqrt{5} - \sqrt{2}$

Hope it was something like this that you were looking for, if there is anything more that you don't understand please post back