Question

Simplify the following :

`(3sqrt(2))/(sqrt(6) - sqrt(3)) - (4sqrt(3))/(sqrt(6) - sqrt(2)) + (2sqrt(3))/(sqrt(6) + 2)`

Answer 1

`(3sqrt(2))/(sqrt(6) - sqrt(3)) - (4sqrt(3))/(sqrt(6) - sqrt(2)) + (2sqrt(3))/(sqrt(6) + 2)`

Rationalizing the denominator of each term, we have

= `(3sqrt(2)(sqrt(6) + sqrt(3)))/((sqrt(6) - sqrt(3))(sqrt(6) + sqrt(3))) - (4sqrt(3)(sqrt(6) + sqrt(2)))/((sqrt(6) - sqrt(2))(sqrt(6) + sqrt(2))) + (2sqrt(3)(sqrt(6) - 2))/((sqrt(6) + 2)(sqrt(6) - 2))`

= `(3sqrt(12) + 3sqrt(6))/(6 - 3) - (4sqrt(18) + 4sqrt(6))/(6 - 2) + (2sqrt(18) - 4sqrt(3))/(2)`

= `(3sqrt(12) + 3sqrt(6))/(3) - (4sqrt(18) + 4sqrt(6))/(4) + (2sqrt(18) - 4sqrt(3))/(2)`

= `sqrt(12) + sqrt(6) - sqrt(18) - sqrt(6) + sqrt(18) - 2sqrt(3)`

= `sqrt(12) - 2sqrt(3)`
= `2sqrt(3) - 2sqrt(3)`
= 0