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Question
If `x = (2sqrt(6))/(sqrt(3) + sqrt(2))`, then what is the value of `(x + sqrt(2))/(x - sqrt(2)) + (x + sqrt(3))/(x - sqrt(3))`?
Options
`sqrt(2)`
`sqrt(3)`
`sqrt(6)`
2
Solution
2
Explanation:
`x = (2sqrt(3) xx sqrt(2))/(sqrt(3) + sqrt(2))`
⇒ `x/sqrt(2) = (2sqrt(3))/(sqrt(3) + sqrt(2))`
⇒ `(x + sqrt(2))/(x - sqrt(2)) = (2sqrt(3) + sqrt(3) + sqrt(2))/(2sqrt(3) - sqrt(3) - sqrt(2))`
= `(3sqrt(3) + sqrt(2))/(sqrt(3) - sqrt(2))`
By componendo and dividendo Similarly,
`x/sqrt(3) = (2sqrt(2))/(sqrt(3) + sqrt(2))`
⇒ `(x + sqrt(3))/(x - sqrt(3)) = (2sqrt(2) + sqrt(3) + sqrt(2))/(2sqrt(2) - sqrt(3) - sqrt(2))`
= `(sqrt(3) + 3sqrt(2))/(sqrt(2) - sqrt(3))`
∴ Expression = `(x + sqrt(2))/(x - sqrt(2)) + (x + sqrt(3))/(x - sqrt(3))`
= `(3sqrt(3) + sqrt(2))/(sqrt(3) - sqrt(2)) + (sqrt(3) + 3sqrt(2))/(sqrt(2) - sqrt(3))`
= `(3sqrt(3) + sqrt(2))/(sqrt(3) - sqrt(2)) + (sqrt(3) + 3sqrt(2))/(sqrt(3) - sqrt(2))`
= `(3sqrt(3) + sqrt(2) - sqrt(3) - 3sqrt(2))/(sqrt(3) - sqrt(2))`
= `(2(sqrt(3) - sqrt(2)))/(sqrt(3) - sqrt(2))`
= 2