Simplification of $\frac{\sqrt{n}}{\sqrt{4n+\sqrt{n}} + 2\sqrt{n}}$

Question

Can somebody explain how pass to this : $\sqrt{n} / \left(\sqrt{4n+\sqrt{n}} + 2*\sqrt{n}\right)$

to this : $1 \big/ \left(2 + 2\sqrt{1+ \frac{\sqrt{n}}{4n}}\right)$

Thanks in advance

Answer 1

\begin{align} \frac{\sqrt{n} }{ \sqrt{4n+\sqrt{n}} + 2\sqrt{n}} =&\;\frac{\frac{\sqrt{n}}{\sqrt{n}} }{ \frac{\sqrt{4n+\sqrt{n}}}{\sqrt{n}} + 2\frac{\sqrt{n}}{\sqrt{n}}}\\[2ex] =&\;\frac{1 }{ \sqrt{\frac{4n+\sqrt{n}}{n}} + 2}\\[2ex] =&\;\frac{1 }{ \sqrt{4+\frac{\sqrt{n}}{n}} + 2}\\[2ex] =&\;\frac{1 }{ \sqrt{4\left(1+\frac{\sqrt{n}}{4n}\right)} + 2}\\[2ex] =&\;\frac{1 }{ 2\sqrt{1+\frac{\sqrt{n}}{4n}} + 2}\\[2ex] =&\;\frac{1}{2 + 2\sqrt{1+ \frac{\sqrt{n}}{4n}}} \end{align}