I'll be working over the base field $\Q$. Let's say an element $\ a\in\C$ is expressible in radicals if the extension $\Q(\a)$ lies within some root tower. That is, $\Q(\a)\s K$, where $K$ lies atop a tower
$$\eq{ \label{tower}\Q\s K_1\s K_2\s \cdots\s K_n = K,\quad K_{i+1}=K_i(d_i) \ \ \text{and}\ \ d_i{}^{n_i}=k_i\in K_i. }$$
Note a polynomial over $\Q$ is solvable in radicals if its root are expressible in radicals.1 Something the reader should note about the definition is that $d_i$ can be chosen to be any of the $n_i$ roots of $k_i$. Because there is another "pop-mathy" definition of expressibility in radicals that says $\a$ can be written as an expression involving whole numbers, the basic arithmetic operations, and $\sqrt[n]{-}$. Here $\sqrt[n]{-}$ refers to the principal root, which is the root with greatest real part out of all roots with nonnegative imaginary parts.
This definition is basically saying that $\Q(\a)\s K$, where $K$ lies atop a tower of the form $\eqref{tower}$, where $K_{i+1}=K_i(d_i)$ with $d_i{}^{n_i}=k_i\in K_i$ and $d_i$ is the principal root $\sqrt[n_i]{k_i}$. Clearly this definition is stronger than the first. To remove ambiguity, let's say $\a$ satisfying the latter definition is expressible in principal roots. Obviously a primitive $n$th root of unity is expressible in radicals, but it is not immediately clear that it is expressible in principal roots---we can't take $\sqrt[n]{1}$ which would just be 1.
Actually the two definitions can be proven to be equivalent, that is expressible in radicals $\imp$ expressible in principal roots. In fact, it clearly suffices to show that every root of unity is expressible in principal roots.
Proof. Let $z_n$ denote a primitive $n$th root of unity. The Galois extension $\Q(z_n)/\Q$ has degree $\varphi(n)$ and is abelian hence solvable, so $\Q(z_n)$ factors as a tower of cyclic extensions. Since $\varphi(n)<n$, by induction we can assume $z=z_{\varphi(n)}$ is expressible in principal roots; adjoin it to every field $F_i$ in the tower. Thus each extension in the tower $F_{i+1}(z)/F_i(z)$ is a radical extension: $F_{i+1}(z) = F_i(z,u_i)$ with $u_i{}^{d_i}\in F_i(z)$ and $d_i\mid\varphi(n)$. Each $u_i$ can be written as a product of some $d_i$th root of unity and a principal root of $F_i$, thus $u_i$ is expressible in principal roots. Thus $z_n\in\Q(z,u_1,\ldots,u_k)$ is expressible in principal roots as desired. $\qed$