NOTE: package and examples notebooks are attached at the end of this post

If you square a surd, and then take the square root, you just get it back. But how do you take the ? if you expand it after you square it?

Sometimes FullSimplify is smart enough:

Usually, it isnt:

Corey Ziegler Hunts wrote a big heuristic package to denest radicals:

(Why Strad?) But usually radicals dont denest:

When they do, the result can be startling:

Can this be right?

Yep. Hopefully this will be obsoleted by Mathematica's developers. But maybe not. I confess near total ignorance of Galois theory, which underlies all the denesting papers. But those papers never exhibit any cool new denestings! SAGE and Maple are rumored to denest, but again, where are goodies like

There are some issues. If there is no denesting, it can take infeasibly long to give up. And it doesn't listify. It misconstrues lists.

Illustrating the widespread underuse of denesting, Theta Function gives:

Tsk.

Is it possible for the nth root of a real binomial to denest to more than five terms? I've never seen one.