Sometimes, we have to rationalize either the numerator or the denominator, and sometimes we can still work the problem without rationalizing. So, in some cases, rationalizing can be done, although it is not necessary, but if it is done, it will be equivalent to the original function, correct? Will it be equivalent to the original function at all points?
This is a very general question: when do we have to rationalize either the numerator or denominator, and when can we still work the problem without doing so? By "general question", I'm interested in various fields of mathematics, not necessarily examples from calculus or algebra. Can you provide some examples/explanations? Thank you!
EDIT: please don't close the question. I'm interested in various explanations to understand better the concept of rationalizing.