`A=1/(\sqrt{1}+\sqrt{2})+1/(\sqrt{2}+\sqrt{3})+1/(\sqrt{3}+\sqrt{4})+...+1/(\sqrt{99}+\sqrt{100})`

`A=(2-1)/(\sqrt{1}+\sqrt{2})+(3-2)/(\sqrt{2}+\sqrt{3})+(4-3)/(\sqrt{3}+\sqrt{4})+...+(100-99)/(\sqrt{99}+\sqrt{100})`

`A=((\sqrt{2}-\sqrt{1})(\sqrt{2}+1))/(\sqrt{1}+\sqrt{2})+((\sqrt{3}-\sqrt{2})(\sqrt{3}+\sqrt{2}))/(\sqrt{2}+\sqrt{3})+((\sqrt{4}-\sqrt{3})(\sqrt{4}+\sqrt{3}))/(\sqrt{3}+\sqrt{4})+...+((\sqrt{100}-\sqrt{99})(\sqrt{100}+\sqrt{99}))/(\sqrt{99}+\sqrt{100})`

`A=\sqrt{2}-\sqrt{1}+\sqrt{3}-\sqrt{2}+\sqrt{4}-\sqrt{3}+...+\sqrt{100}-\sqrt{99}`

`A=-\sqrt{1}+\sqrt{100}`

`A=-1+10`

`A=9`

Vậy `A=9`