156144

Here we have the question,
1/((sqrt1)+(sqrt2))+1/((sqrt2)+(sqrt3))+.............+1/((sqrt99)+(sqrt100))
since we know that (a+b)(a-b)=a^2-b^2
and the easiest method of solving the denominator with surd is multiplying its numerator and the denominator with its conjugate,by doing so at each of the terms we arrive at the form,
{((sqrt1)-(sqrt2))/(1-2)}+{((sqrt2)-(sqrt3))/(2-1)}+..........+{((sqrt99)-(sqrt100))/(99-100)
We can observe that except root 1 and 100,every other numbers have positive nd negative roots with the same denominator -1.
Therefore at last it arrives of form,
{sqrt(1)-sqrt(100)}/-1=90
THEREFORE THE SOLUTION TO THE GIVEN PROBLEM IS 90.