Question

Simplify

`1/(2 + sqrt3) + 2/(sqrt5 - sqrt3) + 1/(2 - sqrt5)`

Answer 1

We know that rationalization factor for `2 + sqrt3`, `sqrt5 - sqrt3` and `2 - sqrt5` are `2 - sqrt3`, `sqrt5 + sqrt3` and `2 + sqrt5` respectively. We will multiply numerator and denominator of the given expression `1/(2 + sqrt3), 2/(sqrt5 - sqrt3) and 1/(2 - sqrt5)` by `2 - sqrt3`, `sqrt5 + sqrt3` and `2 + sqrt5` respectively, to get

`1/(2 + sqrt3) xx (2 - sqrt3)/(2 - sqrt3) + 2/(sqrt5 - sqrt3) xx (sqrt5 + sqrt3)/(sqrt5 + sqrt3) + 1/(2 - sqrt5) xx (2  + sqrt5)/(2 + sqrt5) = (2 - sqrt3)/((2)^2 - (sqrt3)^2) + (2sqrt5 + 2sqrt3)/((sqrt5)^2 - (sqrt3)^2)  + (2 - sqrt5)/((2)^2 - (sqrt5)^2)`

`= (2 - sqrt3)/1 + (2sqrt5 + 2sqrt3)/(5 - 3) + (2 + sqrt5)/(4 - 5)`

`= (2 - sqrt3)/1 + (2sqrt2 + 2sqrt3)/2 + (2 + sqrt5)/(-1)`

`= 2 - sqrt3  + sqrt5 + sqrt3 - sqrt5 - 2`

= 0

Hence the given expression is simplified to 0