Simplifying fraction with nested radicals and fractions

Question

This is my first question here and on a stack exchange in general. I hope my question is precise enough. I have spent a good 15min searching the forum but didn't manage to understand the below.

I am confused by how the fraction to the left of the equal sign is simplified to $\frac{\sqrt 2}{2}$.

$\frac{\sqrt \frac{\frac{a}{2}}{\sqrt 2}}{\sqrt \frac{a}{\sqrt 2}}=\frac{\sqrt 2}{2}$

What steps can I take to get from $\frac{\sqrt \frac{\frac{a}{2}}{\sqrt 2}}{\sqrt \frac{a}{\sqrt 2}}$ to $\frac{\sqrt 2}{2}$ ?

Thanks in advance for your help

Answer 1

$$\frac{\sqrt \frac{\frac{a}{2}}{\sqrt 2}}{\sqrt \frac{a}{\sqrt 2}}=\sqrt{\frac{\frac{a}{2\sqrt2}}{\frac{a}{\sqrt2}}}=\sqrt{\frac{a}{2\sqrt2}\cdot\frac{\sqrt2}{a}}=\sqrt{\frac{1}{2}}=\sqrt{\frac{2}{4}}=\frac{\sqrt2}{2}.$$

Answer 2

$\dfrac{\sqrt \frac{\frac{a}{2}}{\sqrt 2}}{\sqrt \frac{a}{\sqrt 2}} = \sqrt {\dfrac{ \frac{\frac{a}{2}}{\sqrt 2}}{\frac{a}{\sqrt 2}}}$

Go about simplifying $\dfrac{ \frac{\frac{a}{2}}{\sqrt 2}}{\frac{a}{\sqrt 2}}$ and worry about the outside radical at the end.

$\dfrac{ \frac{\frac{a}{2}}{\sqrt 2}}{\frac{a}{\sqrt 2}} = \dfrac{\frac{a}{2}}{a} = \frac 12$

Now apply the outside radical $\sqrt {\frac 12 } = \frac {\sqrt 2}{2}$

Answer 3

$$\frac{\sqrt{\frac{\cfrac{a}{2}}{\,\sqrt 2}\,}}{\sqrt{\cfrac{a}{\sqrt 2}}}= \frac{\sqrt{\cfrac{a}{\,\sqrt 2\,}\cfrac1{2}}}{\sqrt{\cfrac{a}{\sqrt 2}}}=\sqrt{\frac12}= \frac{\sqrt 2}2.$$