Notation. By $\mathbb{Z}_{(2)}$, I mean the localization of $\mathbb{Z}$ at the prime ideal $(2).$ So basically, this is obtained by adjoining a multiplicative inverse for every positive prime number distinct from $2$. Equivalently, we can think of $\mathbb{Z}_{(2)}$ as the set of rational numbers with odd denominator. So: $\mathbb{Z}_{(2)} \subseteq \mathbb{Q}.$ This is a local ring, of course, and judging by how often my commutative algebra lecturer used the phrase "local ring" during those lectures I mostly didn't understand, that's probably important.
Background. Although localization is usually thought of as an "advanced" topic, despite this, I recently recently that $\mathbb{Z}_{(2)}$ actually arises in high school math. In particular, given $x \in \mathbb{R}_{\neq 0}$ and $q \in \mathbb{Z}_{(2)}$, we can make good sense of the expression $x^q.$ Such expressions make sense, in particular even if $x$ is negative. Of course, we can (and often do) replace $q \in \mathbb{Z}_{(2)}$ with $q \in \mathbb{Q}$ or even $q \in \mathbb{R}$, but notice this forces us to assume that $x$ is positive to compensate. There's also a semiring $\mathbb{N}_{(2)}$ obtained by localizing the natural numbers at the prime ideal $(2)$. More concretely: $\mathbb{N}_{(2)} = \{q \in \mathbb{Z}_{(2)} : q \geq 0\}.$
This gives us three exponentiation functions, all denoted $x,q \mapsto x^q$.
\begin{align*} \mathbb{R} \times \mathbb{N}_{(2)} &\rightarrow \mathbb{R} \\ \mathbb{R}_{\neq 0} \times \mathbb{Z}_{(2)} &\rightarrow \mathbb{R}_{\neq 0} \\ \mathbb{R}_{> 0} \times \mathbb{R}_{\color{white}{(2)}} &\rightarrow \mathbb{R}_{> 0} \end{align*}
Question. I'd like to learn a bit more about the number systems $\mathbb{Z}_{(2)}$ and $\mathbb{N}_{(2)}$ from the point of view of these exponentiation functions. For instance, does the localness of $\mathbb{Z}_{(2)}$ tell us anything important about the function $\mathbb{R}_{\neq 0} \times \mathbb{Z}_{(2)} \rightarrow \mathbb{R}_{\neq 0}$ denoted $x,q \mapsto x^q$? If so, I'd like to know about this. Also, I'm interested in "elementary" things that I can teach to high school students about $\mathbb{Z}_{(2)}$ and $\mathbb{N}_{(2)},$ that do not require a lot of abstract mathematics.
Links or references preferred, but direct explanations and further thoughts are also welcome.