Question

In the following determine rational numbers a and b:

`(sqrt3 - 1)/(sqrt3 + 1) = a - bsqrt3`

Answer 1

We know that rationalization factor for `sqrt3 + 1` is `sqrt3 - 1`. We will multiply numerator and denominator of the given expression `(sqrt3 - 1)/(sqrt3 + 1)` by `sqrt3 - 1` to get

`(sqrt3 - 1)/(sqrt3 + 1) xx (sqrt3 - 1)/(sqrt3 - 1) = ((sqrt3)^2 + (1)^2 - 2 xx sqrt3 xx 1)/((sqrt3)^2 - (1)^2)`

`= (3 + 1 - 2sqrt3)/(3 - 2)`

`= (4 - 2sqrt3`)/2`

`= 2 - sqrt3`

On equating rational and irrational terms, we get 

`a - bsqrt3 = 2 - sqrt3`

`= 2 - 1sqrt3`

Hence we get a = 2, b = 1