Question

In the following determine rational numbers a and b:

`(3 + sqrt2)/(3 - sqrt2) = a + bsqrt2`

Answer 1

We know that rationalization factor for `3 - sqrt2` is `3 + sqrt2`. We will multiply numerator and denominator of the given expression `(3 + sqrt2)/(3 - sqrt2)` by `3 + sqrt2` to get

`(3 + sqrt2)/(3 - sqrt2) xx (3 + sqrt2)/(3 + sqrt2) = ((3)^2 + (sqrt2)^2 + 2 xx 3 sqrt2)/((3)^2 - (sqrt2)^2)`

`= (9 + 2 + 6sqrt2)/(9 - 2)`

` = (11 + 6sqrt2)/7`

`= 11/7 + 6/7 sqrt2`

On equating rational and irrational terms, we get 

`a + bsqrt2 = 11/7 + 6/7 sqrt2`

Hence we get a = 11/7, b =  6/7