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Question
Simplify the following :
`sqrt(6)/(sqrt(2) + sqrt(3)) + (3sqrt(2))/(sqrt(6) + sqrt(3)) - (4sqrt(3))/(sqrt(6) + sqrt(2)`
Solution
`sqrt(6)/(sqrt(2) + sqrt(3)) + (3sqrt(2))/(sqrt(6) + sqrt(3)) - (4sqrt(3))/(sqrt(6) + sqrt(2)`
Rationalizing the denominator of each term, we have
= `(sqrt(6)(sqrt(2) - sqrt(3)))/((sqrt(2) + sqrt(3))(sqrt(2) - sqrt(3))) + (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)))`
= `(sqrt(12) - sqrt(18))/(2 - 3) + (3sqrt(12) - 3sqrt(6))/(6 - 3) - (4sqrt(18) - 4sqrt(6))/(6 - 2)`
= `(sqrt(12) - sqrt(18))/(-1) + (3sqrt(12) - 3sqrt(6))/(3) - (4sqrt(18) - 4sqrt(6))/(4)`
= `sqrt(18) - sqrt(12) + sqrt(12) - sqrt(6) - sqrt(18) + sqrt(6)`
= 0
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