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Question
Simplify the following :
`(3sqrt(2))/(sqrt(6) - sqrt(3)) - (4sqrt(3))/(sqrt(6) - sqrt(2)) + (2sqrt(3))/(sqrt(6) + 2)`
Solution
`(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
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