Question

Find the value of a and b in the following:

`(sqrt(2) + sqrt(3))/(3sqrt(2) - 2sqrt(3)) = 2 - bsqrt(6)`

Answer 1

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`