Why use radical notation instead of rational exponents?

(Sorry about the generation error!)

To attempt to answer the actual question, I think that it’s a largely two-fold consequence of the historical inertia mentioned in the comments by @coffeemath. On the one hand, it’s simple classroom inertia: it’s ‘always’ been done this way, so we do it this way. On the other hand it’s the practical consideration that since the radical notation does survive in real-world use, students need to learn how to deal with it. None of this, however, justifies the requirement that students deal with it directly, rather than by translating it into a less cumbersome, more easily manipulated notation. Indeed, in my view this is a good occasion to make the point that well-chosen notation makes our mathematical lives easier.