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Question
Find the value of a and b in the following:
`(sqrt(2) + sqrt(3))/(3sqrt(2) - 2sqrt(3)) = 2 - bsqrt(6)`
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Solution
We have, `(sqrt(2) + sqrt(3))/(3sqrt(2) - 2sqrt(3)) = 2 - bsqrt(6)`
For rationalising the above equation, we multiply numerator and denominator of LHS by `3sqrt(2) + 2sqrt(3)`, we get
`(sqrt(2) + sqrt(3))/(3sqrt(2) - 2sqrt(3)) xx (3sqrt(2) + 2sqrt(3))/(3sqrt(2) + 2sqrt(3)) = 2 - bsqrt(6)`
⇒ `(sqrt(2)(3sqrt(2) + 2sqrt(3)) + sqrt(3)(3sqrt(2) + 2sqrt(3)))/((3sqrt(2))^2 - (2sqrt(3))^2) = 2 - bsqrt(6)` ...[Using identity, (a – b)(a + b) = a2 – b2]
⇒ `(6 + 2sqrt(6) + 3sqrt(6) + 6)/(18 - 12) = 2 - bsqrt(6)`
⇒ `(12 + 5sqrt(6))/6 = 2 - bsqrt(6)`
⇒ `2 + (5sqrt(6))/6 = 2 - bsqrt(6)`
⇒ `bsqrt(6) = - (5sqrt(6))/6`
∴ `b = -5/6`
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