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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.

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Whenever you have alternate expressions for the same value, the choice depends on what you will do with it subsequently. Most of us would simplify $2+2$ to $4$, but if there is a $-2$ in the rest of the expression it might not be a good thing to do. Similarly, there is a bias against roots in the denominator of a fraction, so $\frac {\sqrt 3-1}2$ is preferred to $\frac 1{\sqrt 3+1}$. If you are in the middle of a problem, use whichever suits your needs going forward. If you are handing in a final answer, I would use the latter.

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