Question

I discovered that \(\sqrt{\dfrac{3+\sqrt{5}}{8}} = \dfrac{1+\sqrt{5}}{4}\)

Can anyone prove me that this is true? I don't see how this is true.

Answer 1

Working with the RHS:
\[\begin{align*}\frac{1+\sqrt5}{4}&=\sqrt\frac{(1+\sqrt5)^2}{16}\\
&=\sqrt{\frac{1+2\sqrt5+5}{16}}\\
&=\sqrt{\frac{6+2\sqrt5}{16}}\\
&=\sqrt{\frac{3+\sqrt5}{8}}
\end{align*}\]