I kind of hate the idea of rationalizing the denominator. It’s like some sort of weird rule, like not wearing white after Labor Day, that doesn’t really serve much of a purpose.

But for us, it also does serve a very pragmatic purpose. Our system for grading short-answer problems is not very good at recognizing when two different representations of the same number should be considered equivalent with respect to evaluating a student’s response to a homework problem. If the problem asks how many widgets there are, then it really doesn’t matter if the student answers 8 or \(2^3\). But if the problem asks the student to evaluate \(2^3\), then 8 is a much better answer than \(2^3\). So our homework system is fairly limited in what it accepts as right answers, and one of its quirks is that radicals should be simplified.

And this works great in algebra class. And pretty well in geometry class.

The problem is when we get to calculus class.

Not meaning to give too much away here, but there is a short-answer problem in the calculus class whose right answer is \(\frac{1}{\sqrt{\pi}}\).

One of the students submitted \(\frac{\sqrt{\pi}}{\pi}\) and insisted that this was the better answer because he had “rationalized” the denominator.