[phs]

Radicals are a "natural" extension in a certain sense, just as subtraction and division are. They invert an algebraic operation we often encounter when trying to solve equations handed to us by e.g. physics. I find it understandable to want to give them a name.

Why not something like "those things we can solve with Newton"? As you note Newton is broadly applicable; one would hope, given how popular the need to invert an exponent is, that something better (faster, more stable) if more specific than Newton might be created. It is hard to study a desired hypothetical operation without giving it a name.

On a related note, how come we don't all already have the names of the 4th order iterative operation (iterated exponents) and its inverse in our heads? Don't they deserve consideration? Perhaps, but nature doesn't seem to hand us instances of those operations very often. We seemingly don't need them to build a bridge or solve some other common practical engineering problem. I imagine that is why they fail to appear in high school algebra courses.