The great sum

This is a classic example from Telescoping Series problem.

It can be easily seen that the given expression can be quite simplified if we multiply numerator and denominator by $\sqrt[4]{n+1} - \sqrt[4]{n}$ .

Doing this we get:- $\sum_{n=1}^{9999} \dfrac{ \sqrt[4]{n+1} - \sqrt[4]{n}}{(\sqrt[4]{n+1} - \sqrt[4]{n})(\sqrt[4]{n+1}+\sqrt[4]{n})(\sqrt{n+1}+\sqrt{n})}$ $=\sum_{n=1}^{9999} \dfrac{\sqrt[4]{n+1}-\sqrt[4]{n}}{(\sqrt{n+1}-\sqrt{n})(\sqrt{n+1}+\sqrt{n})}$ $=\sum_{n=1}^{9999} \dfrac{\sqrt[4]{n+1}-\sqrt[4]{n}}{n+1-n}$ $= \sum_{n=1}^{9999} \sqrt[4]{n+1} -\sqrt[4]{n}$ $= \sqrt[4]{2}-1+\sqrt[4]{3} - \sqrt[4]{2}+ \ldots \sqrt[4]{10000} - \sqrt[4]{9999}$ $= -1+10$ $\boxed{=9}$