The sum

\[\displaystyle \frac{1}{2\sqrt{1}+1\sqrt{2}}+\frac{1}{3\sqrt{2}+2\sqrt{3}}+\frac{1}{4\sqrt{3}+3\sqrt{4}}+\dots+\frac{1}{100\sqrt{99}+99\sqrt{100}}\]

can be expressed as \(\frac{a}{b},\) where \(a\) and \(b\) are coprime positive integers.

What is the value of \(a+b\)?

This problem is adapted from a past year KMC Contest.

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