Intersection of two verbal subgroups of abelian group is verbal in both and in whole group

Statement
Suppose $$G$$ is an abelian group, and $$H$$ and $$K$$ are verbal subgroups of $$G$$. Then, $$H \cap K$$ is a verbal subgroup of $$H$$, $$K$$, and $$G$$.

More specifically, if $$H$$ is the subgroup of $$G$$ that is precisely the set of $$m^{th}$$ powers in $$G$$, and $$K$$ is the subgroup of $$G$$ that is precisely the set of $$n^{th}$$ powers in $$G$$. Then, $$H \cap K$$ is:


 * The set of $$l^{th}$$ powers of $$G$$, where $$l$$ is the least common multiple of $$m$$ and $$n$$.
 * The set of $$(l/m)^{th}$$ powers of $$H$$.
 * The set of $$(l/n)^{th}$$ powers of $$K$$.