I'm assuming that you stay within the field of real numbers. You can represent an algebraic real number $\alpha$ by its minimal polynomial $P_\alpha$ plus an additional data, such as the position of $\alpha$ in the set of real roots of $P_\alpha$ (I think there are more efficient encodings). In any case, the main point is that all the usual operations on real algebraic numbers, including radicals, can be performed exactly using the previous description. In particular, given an expression involving radicals, you can compute its minimal polynomial, and you just need to look whether this polynomial has degree 1.

One reference is the book of Bochnak, Coste and Roy, Real algebraic geometry (there may be other but I don't know them). The procedure I outlined here is a very special case of quantifier elimination over real closed fields (for the field of real algebraic numbers).

I'm assuming that you stay within the field of real numbers. You can represent an algebraic real number $\alpha$ by its minimal polynomial $P_\alpha$ plus an additional data, such as the position of $\alpha$ in the set of real roots of $P_\alpha$ (I think there are more efficient encodings). In any case, the main point is that all the usual operations on real algebraic numbers, including real radicals, can be performed exactly using the previous description. In particular, given an expression involving nested real radicals, you can compute its minimal polynomial, and then just need to look whether this polynomial has degree 1.

One reference is the book of Bochnak, Coste and Roy, Real algebraic geometry (there may be other but I don't know them). The procedure I outlined here is a very special case of quantifier elimination over real closed fields (for the field of real algebraic numbers).