Question

In the following, find the values of a and b:

`(sqrt(2) + sqrt(3))/(3sqrt(2) - 2sqrt(3)) = "a" - "b"sqrt(6)`

Answer 1

`(sqrt(2) + sqrt(3))/(3sqrt(2) - 2sqrt(3)`

= `(sqrt(2) + sqrt(3))/(3sqrt(2) - 2sqrt(3)) xx (3sqrt(2) + 2sqrt(3))/(3sqrt(2) + 2sqrt(3)`

= `((sqrt(2) + sqrt(3))(3sqrt(2) + 2sqrt(3)))/((3sqrt(2))^2 - (2sqrt(3))^2`

= `(sqrt(2)(3sqrt(2) + 2sqrt(3)) + sqrt(3)(3sqrt(2) + 2sqrt(3)))/((9 xx 2) - (4 xx 3))`

= `((3 xx 2 + 2sqrt(6)) + (3sqrt(6) + 2 xx 3))/(18 - 12)`

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

= `(12 + 5sqrt(6))/(6)`

= `2 - (-5/6)sqrt(6)`

= `"a" - "b"sqrt(6)`

Hence, a = 2 and b = `-(5)/(6)`.