ToRadicals[expr] attempts to express all Root objects in expr in terms of radicals.
ToRadicals can always give expressions in terms of radicals when the highest degree of the polynomial that appears in any Root object is four.
There are some cases in which expressions involving radicals can in principle be given, but ToRadicals cannot find them.
See The Mathematica Book: Section 3.4.3.
See also: Solve, NSolve, RootReduce, Roots.