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$$\begin{array}{rl} \frac{1}{\sqrt{n}+\sqrt{n+1}} &=\frac{1}{\sqrt{n}+\sqrt{n+1}}\times\frac{\sqrt{n}-\sqrt{n+1}}{\sqrt{n}-\sqrt{n+1}}\\ &=\frac{\sqrt{n}-\sqrt{n+1}}{n-(n+1)}\\ &=\sqrt{n+1}-\sqrt{n}\end{array}$$ $$\begin{array}{rl} \frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+...+\frac{1}{\sqrt{15}+\sqrt{16}} &=(\sqrt{2}-\sqrt{1})+(\sqrt{3}-\sqrt{2})+\\&...+(\sqrt{15}-\sqrt{14})+(\sqrt{16}-\sqrt{15})\\ &=-\sqrt{1}+\sqrt{16}\\ &=3 \end{array} $$