The Root of the Problem

Stage: 4 and 5 Challenge Level: Challenge Level:2 Challenge Level:2

Find the sum of $$\frac{1}{\sqrt{1}+ \sqrt{2}}+ \frac{1}{\sqrt{2}+ \sqrt{3}} + \text{and so on up to}+\frac{1}{ \sqrt {99}+ \sqrt{100}}.$$

You don't need to have learned about summing series to do this problem!
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Published March 2003,April 2003.