<s>[INST] 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?
$$ \frac{\sqrt n}{\sqrt{n + \sqrt{n + \sqrt n}}} = \frac{1}{\sqrt{1 + \sqrt{\frac{1}{n} + \sqrt{\frac{1}{n^3}}}}}$$
 [/INST] Slightly modifying the answer from @evinda:
$\begin{align}
\frac{\frac{\sqrt{n}}{ \sqrt{n}}}{\frac{\sqrt{n+\sqrt{n+\sqrt{n}}}}{\sqrt{n}}}&=\frac{1}{\sqrt{\frac{n+\sqrt{n+\sqrt{n}}}{n}}} \\
&=\frac{1}{\sqrt{1+ \frac{\sqrt{n+\sqrt{n}}}{n}}} \\
&=\frac{1}{\sqrt{1+ \sqrt{\frac{n+\sqrt{n}}{n^2}}}} \\
&=\frac{1}{\sqrt{1+ \sqrt{\frac1n+\frac{\sqrt{n}}{n^2}}}} \\
&=\frac{1}{\sqrt{1+ \sqrt{\frac1n+\sqrt{\frac{n}{n^4}}}}} \\
&=\frac{1}{\sqrt{1+ \sqrt{\frac1n+\sqrt{\frac{1}{n^3}}}}}
\end{align}$
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