Question

How do you rationalize the denominator `1/ (1+ sqrt3 - sqrt5)`?

Answer 1

`1/(1+sqrt(3)-sqrt(5))`

The rationalization of this denominator is a two-set process.
First rationalize the `sqrt(5)`
and then, after some simplification
rationalize the `sqrt(3)`

Let `a=(1+sqrt(3))`
So our initial stage is to rationalize the denominator of
`1/(a-sqrt(5))`

As usual to do this we multiply the numerator and the denominator by the conjugate of the denominator:
`1/(a-sqrt(5)) * (a+sqrt(5))/(a+sqrt(5))`

`= (a+sqrt(5))/(a^2 - 5)`

Substituting `(1+sqrt(3))` back in to this expression in place of `a`.
we get
`= (1+sqrt(3)+sqrt(5))/((1+sqrt(3))^2-5)`

`=(1+sqrt(3)+sqrt(5))/(1 + 2sqrt(3) +3 -5)`

`=(1+sqrt(3)+sqrt(5))/ (2sqrt(3)-1)`

Rationalizing this denominator (using the conjugate of `2sqrt(3)-1`
results in
`((1+sqrt(3)+sqrt(5)))/((2sqrt(3)-1)) * ((2sqrt(3)+1))/((2sqrt(3)+1))`

`=(7+3sqrt(3)+sqrt(5)+2sqrt(15))/11`