#### Question

Simplify `(3sqrt2 - 2sqrt3)/(3sqrt2 + 2sqrt3) + sqrt12/(sqrt3 - sqrt2)`

#### Solution

We know that rationalization factor for `3sqrt2 + 2sqrt3` and `sqrt3 - sqrt2` are `3sqrt2 - 2sqrt3` and `sqrt3 + sqrt2`respectively. We will multiply numerator and denominator of the given expression `(3sqrt2 - 2sqrt3)/(3sqrt2 + 2sqrt3)` and `sqrt12/(sqrt3 - sqrt2)` by `3sqrt2 - 2sqrt3` and `sqrt3 + sqrt2` respectively to get

`(3sqrt2 - 2sqrt3)/(3sqrt2 + 2sqrt3) xx (3sqrt2 - 2sqrt3)/(3sqrt2 - 2sqrt3) + sqrt12/(sqrt3 - sqrt2) xx (sqrt3 + sqrt2)/(sqrt3 + sqrt2) = ((3sqrt2)^2 + (2sqrt3)^2 - 2 xx 3sqrt2 xx 2sqrt3)/((3sqrt2)^2 - (2sqrt3)^2) + (sqrt36 + sqrt24)/((sqrt3)^2 - (sqrt2)^2)`

`= (18 + 12 - 12sqrt6)/(18 - 12) + (6 + sqrt24)/(3 - 2)`

`= (30 - 12sqrt6 + 36 + 12sqrt6)/6`

`= 66/6`

= 11

Hence the given expression is simplified to 11