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Question
If x = `(4sqrt(6))/(sqrt(2) + sqrt(3)` find the value of `(x + 2sqrt(2))/(x - 2sqrt(2)) + (x + 2sqrt(3))/(x - 2sqrt(3)`
Solution
x = `(4sqrt(6))/(sqrt(2) + sqrt(3)`
⇒ `(4sqrt(2) xx sqrt(3))/(sqrt(2) + sqrt(3)`
`x/(2sqrt(2)) = (2sqrt(3))/(sqrt(2) + sqrt(3)`
Applying componendo and dividendo,
`(x + 2sqrt(2))/(x - 2sqrt(2))`
= `(2sqrt(3) + sqrt(2) + sqrt(3))/(2sqrt(3) - sqrt(2) - sqrt(3))`
= `(3sqrt(3) + sqrt(2))/(sqrt(3) - sqrt(2)` ...(i)
Again `x/(2sqrt(3)) = (2sqrt(2))/(sqrt(2) + sqrt(3)`
Applying componendo and dividendo,
`(x + 2sqrt(3))/(x - 2sqrt(3))`
= `(2sqrt(2) + sqrt(2) + sqrt(3))/(2sqrt(2) - sqrt(2) - sqrt(3))`
= `(3sqrt(2) + sqrt(3))/(sqrt(2) - sqrt(3)` ...(ii)
Adding (i) and (ii)
`(x + 2sqrt(2))/(x - 2sqrt(2)) + (x + 2sqrt(3))/(x - 2sqrt(3)`
= `(3sqrt(3) + sqrt(2))/(sqrt(3) - sqrt(2)) + (3sqrt(2) + sqrt(3))/(sqrt(2) - sqrt(3)`
= `(3sqrt(3) + sqrt(2))/(sqrt(3) - sqrt(2)) - (3sqrt(2) + sqrt(3))/(sqrt(3) - sqrt(2)`
= `(3sqrt(3) + sqrt(2) - 3sqrt(2) - sqrt(3))/(sqrt(3) - sqrt(2)`
= `(2sqrt(3) - 2sqrt(2))/(sqrt(3) - sqrt(2)`
= `(2(sqrt(3) - sqrt(2)))/(sqrt(3) - sqrt(2)`
= 2.
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