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?
$$ \frac{\sqrt n}{\sqrt{n + \sqrt{n + \sqrt n}}} = \frac{1}{\sqrt{1 + \sqrt{\frac{1}{n} + \sqrt{\frac{1}{n^3}}}}}$$

Suppose that $n>0$.
$$\frac{\sqrt{n}}{ \sqrt{n+\sqrt{n+\sqrt{n}}} } = \frac{1}{ \frac{1}{\sqrt{n}} \left( \sqrt{n+\sqrt{n+\sqrt{n}}} \right) } = \frac{1}{ \sqrt{1+ \frac{1}{n} \left( \sqrt{n + \sqrt{n}} \right) } } = \frac{1}{\sqrt{1+ \sqrt{ \frac{1}{n} + \frac{1}{n^2} \sqrt{n} } } } = \frac{1}{\sqrt{1+ \sqrt{ \frac{1}{n} + \sqrt{ \frac{1}{n^3}} } } } $$.
