Why do math teachers stress the process of rationalizing numerator and denominator?
This is more a problem, I see, in terms of rationalizing the denominator.
Why can't fractions have a radical in the denominator? What's the big deal?
It is my opinion that conventions in rationalising, especially avoiding radicals in the denominator originated in a time when typesetting and printing technologies were considerably less advanced than they are today. Conventions grew up to make the typesetting easier.
Nowadays, I would encourage people to select the form that most clearly expresses the point they wish to make, the meaning they wish to convey, or the value the expression has.
If you are doing lots of compicklated mathematics and you end up with lots of messy irrational algebraic numbers at the end of it, it makes jolly good sense to convert them all into some sort of canonical form, otherwise you may find yourself not recognising that two numbers that look different are in fact the same.
Srslydudz, canonical forms of all sorts of objects are useful things to have around the shop. If it's difficult and hard work to get to them, then you need more practice.