Question

Simplify the following :

`(4sqrt(3))/((2 - sqrt(2))) - (30)/((4sqrt(3) - 3sqrt(2))) - (3sqrt(2))/((3 + 2sqrt(3))`

Answer 1

`(4sqrt(3))/((2 - sqrt(2))) - (30)/((4sqrt(3) - 3sqrt(2))) - (3sqrt(2))/((3 + 2sqrt(3))`

Rationalizing the denominator of each term, we have

= `(4sqrt(3)(2 + sqrt(2)))/((2 - sqrt(2))(2 + sqrt(2))) - (30(4sqrt(3) + 3sqrt(2)))/((4sqrt(3) - 3sqrt(2))(4sqrt(3) + 3sqrt(2))) - (3sqrt(2)(3 - 2sqrt(3)))/((3 + 2sqrt(3))(3 - 2sqrt(3))`

= `(8sqrt(3) + 4sqrt(6))/(4 - 2) - (120sqrt(3) + 90sqrt(2))/(48 - 18) - (9sqrt(2) - 6sqrt(6))/(9 - 12)`

= `(8sqrt(3) + 4sqrt(6))/(2) - (120sqrt(3) + 90sqrt(2))/(30) - (9sqrt(2) - 6sqrt(6))/(-3)`

= `(8sqrt(3) + 4sqrt(6))/(2) - (120sqrt(3) + 90sqrt(2))/(30) - (9sqrt(2) - 6sqrt(6))/(3)`

= `4sqrt(3) + 2sqrt(6) - 4sqrt(3) - 3sqrt(2) + 3sqrt(2) - 2sqrt(6)`
= 0