Socle is powering-invariant in solvable group

Statement
Suppose $$G$$ is a solvable group. Then, the socle $$\operatorname{Soc}(G)$$, defined as the subgroup generated by all the minimal normal subgroups of $$G$$, is a powering-invariant subgroup of $$G$$.

Facts used

 * 1) uses::Minimal normal implies powering-invariant in solvable group

Proof
The proof follows from Fact (1), and the observation that since all the minimal normal subgroups pairwise centralize each other, their join, the socle, is also powering-invariant.