Divisible not implies rationally powered

Statement
It is possible for a divisible group (i.e., a group in which every element has a $$n^{th}$$ root for every $$n$$) to not be a rationally powered group (i.e., there is at least one element and one $$n$$ for which the $$n^{th}$$ root is not unique).

Proof
The simplest example is the group of rational numbers modulo integers $$\mathbb{Q}/\mathbb{Z}$$. We can also take the circle group $$\R/\mathbb{Z}$$. Also, any general linear group over the field of complex numbers (or in general, over any algebraically closed field of characteristic zero) satisfies the condition.