ABC26GN1310 · Square Roots and Cube Roots

Given expression $$\begin{aligned} & =\frac{1}{\sqrt{100}-\sqrt{99}} \times \frac{\sqrt{100}+\sqrt{99}}{\sqrt{100}+\sqrt{99}}-\frac{1}{\sqrt{99}-\sqrt{98}} \times \frac{\sqrt{99}+\sqrt{98}}{\sqrt{99}+\sqrt{98}}+\frac{1}{\sqrt{98}-\sqrt{97}} \times \frac{\sqrt{98}+\sqrt{97}}{\sqrt{98}+\sqrt{97}} \\ & \quad-\frac{1}{\sqrt{97}-\sqrt{96}} \times \frac{\sqrt{97}+\sqrt{96}}{\sqrt{97}+\sqrt{96}}+\ldots+\frac{1}{\sqrt{2}-\sqrt{1}} \times \frac{\sqrt{2}+\sqrt{1}}{\sqrt{2}+\sqrt{1}} \\ & =\frac{\sqrt{100}+\sqrt{99}}{(100-99)}-\frac{\sqrt{99}+\sqrt{98}}{(99-98)}+\frac{\sqrt{98}+\sqrt{97}}{(98-97)}-\frac{\sqrt{97}+\sqrt{96}}{(97-96)}+\ldots+\frac{\sqrt{2}+\sqrt{1}}{(2-1)} \\ & =(\sqrt{100}+\sqrt{99})-(\sqrt{99}+\sqrt{98})+(\sqrt{98}+\sqrt{97})-(\sqrt{97}+\sqrt{96})+\ldots+(\sqrt{2}+\sqrt{1}) \\ & =\sqrt{100}+\sqrt{1}=10+1=11 . \end{aligned}$$