Is there a symbol for plus and minus as opposed to plus or minus? I know that you can use $\pm$ for when the answer could be either positive or negative, e.g., $x^2=16$, $x=\pm 4$.
But is there a symbol that implies that you use both the positive and the negative values?  For example, I want to do something along the lines of:
$$(2/3a) \left(\sqrt[3]{2b^3 - 9abc + \sqrt{−4(b^2−3a)}} + \sqrt[3]{2b^3 - 9abc - \sqrt{−4(b^2−3a)}}\right)$$
It would be very useful to not have to write out the cube root twice and instead have a plus and minus sign before the square root.
 A: Another way to write it is:
$$\frac{2}{3a}\left(R_+ + R_-\right) \text{ where } R_\pm = \sqrt[3]{2b^3-9abc\pm\sqrt{-4(b^2-3a)}}$$
A: There's no single symbol for it. So just use, $$\sum_{n=0}^1\sqrt[3]{2b^3-9abc+(-1)^n\sqrt{-4(b^2-3a)}}$$
A: No, and there's a good reason for it: it cannot convey the necessary information.
How would the reader know that the intention is to add?
What if you intended the "and" to be for multiplication?
You need to denote the operator somehow, and that will take care of the "and" part by itself.
A: You can do this with multisets:
$$\sum(2×(a \pm b))$$
is the expression you want.
Basically, $\pm b$ is the multiset $\{b,-b\}$, so $a \pm b$ is $\{a+b,a-b\}$, so $2 \times (a \pm b)$ is the multiset $\{2(a+b),2(a-b)\}$, then the $\sum$ just means "take the sum."
Edit. Let me add that this notation, being not entirely standard, should never be used without prior explanation. Putting all that side, however, I want to draw attention to a cultural problem here. That problem, in short, is a general aversion to new and interesting ways of denoting our thoughts and ideas. This aversion is holding mathematics back; see here, for example. Here's how it plays out in practice.
What should happen.
The reader encounters a new and unfamiliar notation. Suddenly intrigued by the possibility of denoting his ideas more tersely and clearly, he does his best to understand the conventions of that notation and/or any deeper ideas on which it is predicated, and spends some time experimenting with it. He writes a few proofs in the new and unfamiliar language. After the new language has become sufficiently familiar, he makes a judgement regarding whether or not the benefits of the new notation outweigh the costs, and makes a conscious decision to either adopt it in his own work and writing, or not to adopt it. If he chooses the former, he is unconcerned about the potential reduction in readership, because he knows that truly elite mathematicians are intellectually flexible, and that in fact, most people are quite flexible once they've adopted the right mindset. He therefore knows that, by adopting the best possible conventions that he can, he is fundamentally doing other people a favor, and that and while this may infuriate some, nonetheless the benefits of adopting the best possible conventions outweigh the costs, and that is that.
What tends to happen.
The reader encounters a new and unfamiliar notation. The parts of his brain that are responsible for tribal thinking instantly categorize the person using the notation as an "outsider" whose opinions and ideas fundamentally don't matter (unless they're already high-up in the mathematical community, in which its an automatic movement to "what should happen"). He thinks to himself: how dare she write this kind of drivel? He starts to feel angry, and he is convinced that his anger is rational and justified. Already, whether or not the new notation could be useful to him - or to mathematics - has ceased to matter. Non-standard notation! he growls. The tools of cowards! The fact that the writer, in all likelihood, is using this notation precisely because she found it to be useful has ceased to matter. All that matters is how best to attack this new and unfamiliar experience. After composing himself for a moment, he decides on the lecturing approach. He will simply talk down to the other person, until she finally "gets" that pandering to people's inflexibility is a good and noble pursuit. He leaves a comment to the effect that: "Look, if you don't want to reach the largest possible audience, then keep on writing that way." The OP reads this comment with great sadness. They could have learned something, she thinks to herself. Instead, they have learned nothing.
Take home message.
Attitudes that justify notational or intellectual stagnation deserve to be regarded with both suspicion and sadness (in perhaps equal parts), especially when they're predicated on tribal thinking in which the ingroup is privileged and ideas from the outgroup ignored. We should be careful to reject those attitudes that, by retarding progress, make mathematics worse, while consciously adopting beliefs that make it better.
