## Solution

Can you rationalise the denominator of the following fractions?

1. $\dfrac{1}{1+\sqrt{2}}$
• Which of these approaches do you prefer?

• Which of these approaches can be most easily adapted for the other problems or other situations?

For the remaining questions, we will only offer one approach to solving them, but as we have seen, there are several other possibilities.

1. $\dfrac{1}{2+\sqrt{2}}$
1. $\dfrac{1}{\sqrt{2} + \sqrt{4}}$
1. $\dfrac{1}{1 - \sqrt{2} + \sqrt{4}}$

#### Taking it further: What difference does a sign make?

1. $\dfrac{1}{1 + \sqrt{2} + \sqrt{4}}$
1. $\dfrac{1}{1 + \sqrt{2} - \sqrt{4}}$

The point of these problems is not so much to work out the answers to these particular questions, but rather to show that we can, in principle, work out the answers. Computer algebra systems which do these calculations use techniques such as these behind the scenes to do their work.

Can you find any fractions for which you cannot rationalise the denominator?

The approach we have taken has involved finding a polynomial with the denominator as a root. There are numbers which are not the root of any polynomial with integer coefficients, such as $1+\pi$ (see Squaring the circle), so a fraction such as $\dfrac{1}{1+\pi}$ cannot have its denominator rationalised.