156012

the ques asks for x,
x=1/(sqrt1+sqrt2) +1/(sqrt2+sqrt3) + . . . + 1/(sqrt99+sqrt100)
now rationalizing dwnominators of all terms ,
i.e multiplying them by ther conjugates(mid sign changed) to get a rational no below,
eg 1/(sqrta+sqrtb)=(sqrta-sqrtb)/[(sqrta+sqrtb)(sqrta-sqrtb)]
=(sqrta-sqrtb)/(a-b)
from identity (x+y)(x-y)=-x^2-y^2
so we get
x= (sqrt2-sqrt1)/(2-1) +(sqrt3-sqrt2)/(3-2) + . . . +(sqrt100-sqrt99)/(100-99)
now if we notice, all denominators are differences between consecutive integers and hence=1
2-1=3-2=...=100-99=1
and any number divided by one is the number itself ,
so , x= sqrt2-sqrt1+sqrt3-sqrt2+...+sqrt99-sqrt98+sqrt100-sqrt99
now all terms except -sqrt1 and sqrt100 get cancelled
thus x = sqrt100-sqrt1=10-1=9