Question

In the following determine rational numbers a and b:

`(5 + 3sqrt3)/(7 + 4sqrt3) = a + bsqrt3`

Answer 1

We know that rationalization factor for `7 + 4sqrt3` is `7 - 4sqrt3`. We will multiply numerator and denominator of the given expression `(5 + 3sqrt3)/(7 + 4sqrt3)`  by `7 - 4sqrt3` to get

`(5 + 3sqrt3)/(7 + 4sqrt3) xx (7 - 4sqrt3)/(7 - 4sqrt3) = (5 xx 7 - 5 xx 4 xx sqrt3 + 3 xx 7 xx sqrt3 - 3 xx 4 xx (sqrt3)^2)/((7)^2 - (4sqrt3)^2)`

`= (35 - 20sqrt3 + 21sqrt3 - 36)/(49 - 49)`

`= (sqrt3 - 1)/1`

`= sqrt3 - 1`

On equating rational and irrational terms, we get 

`a + bsqrt3 = sqrt3 - 1`

`= -1 + 1sqrt3`

Hence we get a = -1, b = 1