Question

In the following determine rational numbers a and b:

`(sqrt11 - sqrt7)/(sqrt11 + sqrt7) = a - bsqrt77`

Answer 1

 We know that rationalization factor for `sqrt11 + sqrt7` is `sqrt11 - sqrt7`. We will multiply numerator and denominator of the given expression `(sqrt11 - sqrt7)/(sqrt11 + sqrt7)` by  `sqrt11 - sqrt7` to get

`(sqrt11 - sqrt7)/(sqrt11 + sqrt7) xx (sqrt11 - sqrt7)/(sqrt11 - sqrt7) = ((sqrt11)^2 + (sqrt7)^2 - 2 xx sqrt11 xx sqrt7)/(sqrt(11)^2 - sqrt(7)^2)`

`= (11 + 7 - 2 sqrt77)/(11 - 7)`

`= (18 - 2sqrt77)/4`

`= 9/2 - 1/2 sqrt77`

On equating rational and irrational terms, we get 

`a - bsqrt77 = 9/2 - 1/2 sqrt77`

Hence we get a = 9/2, b = 1/2