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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
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