Playing with Square Root

Algebra Level 3

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.