Rationalising Denominators of Multiple Fractions

Worked Example of Rationalising Denominators

Simplify $\displaystyle\frac{1}{\sqrt{1} + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \cdots + \frac{1}{\sqrt{99} + \sqrt{100}}$.

\begin{aligned} \displaystyle \require{color} &= \frac{1}{\sqrt{1} + \sqrt{2}} \times \frac{\sqrt{1}-\sqrt{2}}{\sqrt{1}-\sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} \times \frac{\sqrt{2}-\sqrt{3}}{\sqrt{2}-\sqrt{3}} + \cdots + \frac{1}{\sqrt{99} + \sqrt{100}} \times \frac{\sqrt{99}-\sqrt{100}}{\sqrt{99}-\sqrt{100}} \\ &= \frac{\sqrt{1}-\sqrt{2}}{1-2} + \frac{\sqrt{2}-\sqrt{3}}{2-3} + \cdots + \frac{\sqrt{99}-\sqrt{100}}{99-100} \\ &= -(\sqrt{1}-\sqrt{2})-(\sqrt{2}-\sqrt{3})-\cdots-(\sqrt{99}-\sqrt{100}) \\ &= -\sqrt{1} + \sqrt{2}-\sqrt{2} + \sqrt{3}-\cdots-\sqrt{99} + \sqrt{100} \\ &= -\sqrt{1} + \sqrt{100} \\ &= -1 + 10 \\ &= 9 \end{aligned}

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