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Question
Find the value of
`(x + sqrt(3))/(x - sqrt(3)) + (x + sqrt(2))/(x - sqrt(2)), if x = (2sqrt(6))/(sqrt(3) + sqrt(2)`.
Solution
`(2sqrt(6))/(sqrt(3) + sqrt(2)`
or
x = `(2 xx sqrt(3) xx sqrt(2))/(sqrt(3) + sqrt(2)`
⇒ `x/sqrt(3) = (2sqrt(2))/(sqrt(3) + sqrt(2)`
Applying Componendo and Dividendo
`(x + sqrt(3))/(x - sqrt(3)) = (2sqrt(2) + sqrt(3) + sqrt(2))/(2sqrt(2) - sqrt(3) - sqrt(2)`
= `(3sqrt(2) + sqrt(3))/(sqrt(2) - sqrt(3)`
`(x + sqrt(3))/(x - sqrt(3)) = (3sqrt(2) + sqrt(3))/(-(sqrt(3) - sqrt(2))` ...(i)
Also `x/sqrt(2) = (2sqrt(3))/(sqrt(3) + sqrt(2)`
Applying Componendo and Dividendo
`(x + sqrt(2))/(x + sqrt(2)) = (2sqrt(3) + sqrt(3) + sqrt(2))/(2sqrt(3) - sqrt(3) - sqrt(2)`
`(x + sqrt(2))/(x - sqrt(2)) = (3sqrt(3) + sqrt(2))/(sqrt(3) - sqrt(2)` ...(ii)
Adding (i) and (ii)
`(x + sqrt(3))/(x - sqrt(3)) + (x + sqrt(2))/(x - sqrt(2)) = (-3sqrt(2) - sqrt(3) + 3sqrt(3) + sqrt(2))/(sqrt(3) - sqrt(2)`
= `(-2sqrt(2) + 2sqrt(3))/(sqrt(3) - sqrt(2)`
= `(2(sqrt(3) - sqrt(2)))/((sqrt(3) - sqrt(2))`
= 2.
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