Is there an equivalent of $\pm$ but for $\times$ and $\div$?

Question

Is there mathematical sign that combines multiplication ($\times$) and division ($\div$) operations? Would that ever be used? For example $4({\times}{\div})2$ would give $8$ and $2$, like $4\pm 2$ would give $6$ and $2$.

Answer 1

I've not seen one, but something one can do is write e.g. $ab^{\pm 1}$ for "$a$ times or divided by $b$".

Answer 2

For the multiplicative equivalent of $\pm$, one can write either $ab^{\pm{1}}$ (as in fish's answer) or $e^{\ln a \pm \ln b}$.

Answer 3

Fish's answer, which I upvoted, is the simplest solution. Let me just describe another possible notation, in a real context, to answer twosigma's comment.

Let $A = \{a_1, \ldots, a_k\}$ be a finite set. Then the elements of the free group over $A$ are products of the form $a_{i_1}^{\epsilon_1} \dotsm a_{i_n}^{\epsilon_n}$, where each $\epsilon_j$ is either $1$ or $-1$.