Question

In the following determine rational numbers a and b:

`(4 + 3sqrt5)/(4 - 3sqrt5) = a + bsqrt5`

Answer 1

We know that rationalization factor for `4 - 3sqrt5` is `4 + 3sqrt5`. We will multiply numerator and denominator of the given expression `(4 + 3sqrt5)/(3 - 3sqrt5)` by `4 + 3sqrt5` to get

`(4 + 3sqrt5)/(4 - 3sqrt5) xx (4 + 3sqrt5)/(4 + 3sqrt5) = ((4)^2 + (3sqrt3)^2 + 2 xx 4 xx 3sqrt5)/((4)^2 - (3sqrt5)^2)`

`= (16 + 45 + 24sqrt5)/(16 - 45)`

`= (61 + 24sqrt5)/(-29)`

`= -61/29 - 24/29 sqrt5`

On equating rational and irrational terms, we get 

`a + bsqrt5 = -61/29 - 24/29 sqrt5`

Hence we get `a = -61/29, b = -24/29`