How do you say "ALL the $n$th roots"?

Question

$\sqrt[2]{1}$ is strictly only $1$, despite the equation $x^{2}=1$ having two solutions: $1$ and $-1$. Same with cube roots; $\sqrt[3]{1}$ is strictly just $1$, despite $\left(-\frac{i\sqrt[2]{3}+1}{2}\right)^3$ and $\left(-\frac{i\sqrt[2]{3}-1}{2}\right)^3$ both being $1$ as well. Is there a way to say "ALL the cube roots of 1" or, more broadly, "All the $n$th roots of $x$"?

Answer 1

In informal usage, you can say "the $n$th roots of unity"; that is generally understood to mean all complex $n$th roots of $1$. If you want to emphasize this—say, because you've previously been restricting yourself to real values—you can say "the $n$ complex $n$th roots of unity."

Symbolically, you can write

$$ \{ z \in \mathbb{C} \mid z^n = 1 \} $$

If these don't suffice, you may want to edit your question to clarify what requirements you have for the wording.

Answer 2

The set of all $r$ such that $r^n=x$ can be written as $\{r \in\mathbb{C}: r^n = x\}$ and is equivalent to the set of all the $n$th roots of $x$.