The sum of the expression <span class="math-tex">\(\frac 1 {\sqrt 1 + \sqrt 2} + \frac 1 {\sqrt 2 + \sqrt 3} + \frac 1 {\sqrt 3 + \sqrt 4} + ... + \frac 1 {\sqrt {80} + \sqrt {81}}\)</span> is
\[ \begin{aligned} S &= \frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+...+\frac{1}{\sqrt{80}+\sqrt{81}} \\ &= \frac{1}{\sqrt{1}+\sqrt{2}} \times \frac{\sqrt{2}-\sqrt{1}}{\sqrt{2}-\sqrt{1}} + \frac{1}{\sqrt{2}+\sqrt{3}} \times \frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}-\sqrt{2}} \\ &\quad + \frac{1}{\sqrt{3}+\sqrt{4}} \times \frac{\sqrt{4}-\sqrt{3}}{\sqrt{4}-\sqrt{3}} + ... + \frac{1}{\sqrt{80}+\sqrt{81}} \times \frac{\sqrt{81}-\sqrt{80}}{\sqrt{81}-\sqrt{80}} \\ &= (\sqrt{2}-\sqrt{1}) + (\sqrt{3}-\sqrt{2}) + (\sqrt{4}-\sqrt{3}) + ... + (\sqrt{80}-\sqrt{79}) + (\sqrt{81}-\sqrt{80}) \\ &= \sqrt{81} - \sqrt{1} \\ &= 9 - 1 \\ &= 8 \end{aligned} \]
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