Find the value of: `1/(1+sqrt2)+1/(sqrt2+sqrt3)+1/(sqrt3+sqrt4)+...1/(sqrt99+sqrt100)`

Answer

N/a

Explanation

`(1)/(sqrt(1)+sqrt(2))=(1)/(sqrt(2)+sqrt(1))=(1)/(sqrt(2)+sqrt(1))xx(sqrt(2)-sqrt(1))/(sqrt(2)-sqrt(1))=(sqrt(2)-1)/(2-1)=sqrt(2)-1`
`(1)/(sqrt(2)+sqrt(3))=(1)/(sqrt(3)+sqrt(2))=(1)/(sqrt(3)+sqrt(2))xx(sqrt(3)-sqrt(2))/(sqrt(3)-sqrt(2))=(sqrt(3)-sqrt(2))/(3-2)=sqrt(3)=sqrt(2)`
similarly, `(1)/(sqrt(3)+sqrt(4))=sqrt(4)-sqrt(3), . . . (1)/(sqrt(99)+sqrt(100))=10-sqrt(99)`
`:."Given expression"=(sqrt(2)-1)+(sqrt(3)=sqrt(2))+sqrt(4)-sqrt(3))+ . . . +(10-sqrt(99))`
(rational terms, only from first and last bracket will remain, all others will be cancelled out)
=10-1=9