Dividing by $\sqrt n$ Why is the following equality true? 
I know I should divide by $\sqrt n$ but how is it done exactly to get the RHS?
$$ \frac{\sqrt n}{\sqrt{n + \sqrt{n + \sqrt n}}} = \frac{1}{\sqrt{1 + \sqrt{\frac{1}{n} + \sqrt{\frac{1}{n^3}}}}}$$

Well this is calculus. As you said yourself, you divide the numerator and the denominator by $\sqrt{n}$
for the numerator it's easy, as $\frac{\sqrt{n}}{\sqrt{n}}=1$.
for the denominator you get \begin{equation}
\frac{1}{\sqrt{n}}\sqrt{n+{\sqrt{n+{\sqrt{n}}}}}=\sqrt{1+\frac{\sqrt{n+\sqrt{n}}}{n}}
\end{equation}
Keep in mind, if you want to pull a factor under a square root, you have to square it!
the above can be rewritten as \begin{equation}
\sqrt{1+\sqrt{\frac{n}{n^2}+\frac{\sqrt{n}}{n^2}}}= \sqrt{1+\sqrt{\frac{1}{n}+\sqrt{\frac{n}{n^4}}}}=\sqrt{1+\sqrt{\frac{1}{n}+\sqrt{\frac{1}{n^3}}}}
\end{equation}
And you're done
