Question

In the following determine rational numbers a and b:

`(4 + sqrt2)/(2 + sqrt2) = n - sqrtb`

Answer 1

We know that rationalization factor for `2 + sqrt2` is `2 - sqrt2`. We will multiply numerator and denominator of the given expression `(4 + sqrt2)/(2 + sqrt2)` by `2 - sqrt2` to get

`(4 + sqrt2)/(2 + sqrt2) xx (2 -  sqrt2)/(2 - sqrt2) = (4 xx 2 - 4 xx sqrt2 + 2 xx sqrt2 - (sqrt2)^2)/((2)^2 - (sqrt2)^2)`

`= (8 - 4sqrt2 + 2sqrt2 - 2)/(4 - 2)`

`= (6 - 2sqrt2)/2`

`= 3 - sqrt2`

On equating rational and irrational terms, we get 

`a - sqrtb = 3 - sqrt2`

Hence we get a = 3, b = 2