Expressing $\sqrt{3 − \sqrt{5}}$ in the form $\frac{\sqrt{a}−\sqrt{b}}{c}$

Question

Express $\sqrt{3 − \sqrt{5}}$ in the form $\frac{\sqrt{a}−\sqrt{b}}{c}$ for some integers $a,b,c$.

I'm not quite sure how to start this, can anyone give me a little hint or two

Answer 1

I remember I solved this kind of problem when I was a pupil, a lot of nostalgia :D.

You have
\begin{align} \sqrt{3-\sqrt{5}} &= \frac{\sqrt{6-2\sqrt{5}}}{\sqrt{2}} \\ &= \frac{\sqrt{1-2\sqrt{5} +5}}{\sqrt{2}} \\ &= \frac{\sqrt{(\sqrt{5}-1)^2}}{\sqrt{2}} \\ &= \frac{\sqrt{5}-1}{\sqrt{2}} \\ &= \frac{\sqrt{10}-\sqrt{2}}{2} \\ \end{align}

Answer 2

$\dfrac{3-\sqrt{5}}{2}= \dfrac{6-2\sqrt{5}}{4}= \dfrac{(1-\sqrt{5})^2}{4}\implies \sqrt{\dfrac{3-\sqrt{5}}{2}}=\dfrac{\sqrt{5}-1}{2}\implies \dfrac{\sqrt{3-\sqrt{5}}}{\sqrt{2}}=\dfrac{\sqrt{5}-1}{2}\implies \sqrt{3-\sqrt{5}}=\sqrt{2}\cdot \dfrac{\sqrt{5}-1}{2}=\dfrac{\sqrt{10}-\sqrt{2}}{2}$.