Question

How do you rationalize the denominator `(3 sqrt 5 + 2) / (3 sqrt 5 + 5)`?

Answer 1

You must look for a binomial that multiplied by the denominator makes the root disappear.

Usually, when you want to rationalize a binomial denominator you will multiply it by its conjugate, in this case `(3sqrt(5)-5)`.
As you know, in order to preserve the fraction, you will multiply it by 1, which is the same as `(3sqrt(5)-5)/(3sqrt(5)-5)`.

`((3sqrt(5)+2))/((3sqrt(5)+5))((3sqrt(5)-5))/((3sqrt(5)-5))=(45-15sqrt(5)+6sqrt(5)-10)/(45-25)=(35-11sqrt(5))/20`