General linear group over algebraically closed field is divisible

Statement
Suppose $$K$$ is a field that is algebraically closed and $$G$$ is a general linear group of finite degree $$n$$ over $$K$$, i.e., $$G = GL(n,K)$$. Then, $$G$$ is a divisible group, i.e., for any $$g \in G$$ and any positive integer $$n$$, there exists $$x \in G$$ (not necessarily unique) such that $$x^n = g$$.

Related facts

 * Triangulability theorem

Proof
The idea is to first conjugate to a Jordan canonical form matrix, then take the unique $$n^{th}$$ root of that.