I admit this question is quite general.
If we have a group (or perhaps some other algebraic structure) $G$, we can define the Cartesian product $G\times G$ of $G$ with itself. And then powers of $G$ as $G^{\times m} = G \times G~ \times ...~m$ times$~...\times ~ G$.
In the case above $m$ is required to be a natural number. My question is when are we allowed to let $m$ be negative or rational. In the case of finite Abelian Groups we can obviously factor the cyclic components of a group as we would an integer and hence $(\mathbb{Z}_p \times \mathbb{Z}_p)^{\times 1/2} = \mathbb{Z}_p$ et cetera. But I have no idea what the requirements would be for infinite or non Abelian groups.
This still doesn't address the possibility of negative $m$. I haven't been able to think of anything on this subject, other than how $G^{\times -1}$ shouldn't have to be a group, like how $\frac{1}{n}$ isn't always an integer where $n \in \mathbb{Z}$. It would actually be rather strange if it was a group, because then we would have to worry about how many elements it has. But if not a group, what could it be. Something devilishly complicated no doubt!
Negative $m$ is impossible in a fairly general sense. First, recall that the product of groups is a special case of the categorical product, whose unit is the terminal object $1$.
Theorem: Let $c, d$ be objects in a category $C$ with finite products such that $c \times d \cong 1$. Then $c \cong d \cong 1$.
Proof. If $c \times d \cong 1$, then $\text{Hom}(-, c \times d) \cong \text{Hom}(-, 1) \cong 1 \cong \text{Hom}(-, c) \times \text{Hom}(-, d)$, from which it follows that $\text{Hom}(-, c) \cong \text{Hom}(-, d) \cong 1$, so $c, d$ both satisfy the universal property of the terminal object. $\Box$
So the categorical product can never have nontrivial inverses. Nevertheless, there is sometimes a useful sense in which we can virtually consider the negative of an object of a category; see Grothendieck group.
In the rational case I suspect the answer need not be unique but I don't know an explicit example.