Rationalising a fraction with a surd

Question

The given fraction is: $$\frac{2}{1+\sqrt5}$$

Can someone explain to me how to rationalise this (in steps - GCSE Level)?

My only idea is to mutliply the top and bottom by $1+\sqrt5$ ??

TIA.

Answer 1

$$ \frac 2 {1+ \sqrt 5} = \frac {2 (1-\sqrt 5)}{(1+\sqrt 5)(1 - \sqrt 5)} = \frac{2(1-\sqrt 5)}{1 - 5} = \frac{2(1-\sqrt 5)}{-4} = \frac{-(1-\sqrt 5)}{2} = \frac{\sqrt 5 - 1} 2. $$

Answer 2

When you see one of $a+\sqrt{b}$ or $a-\sqrt{b}$ in the denominator, you need to supply the other one (on top and bottom) so that both forms appear together in the denominator.

The reason is to take advantage of the fact that $$(a+\sqrt{b})(a-\sqrt{b}) = a^2-a\sqrt{b}+a\sqrt{b}-(\sqrt{b})^2 = a^2-b$$ (assuming $b$ is a positive number).