What is $i^{\frac13}$?

Question

When asked the value of $i^{\frac13}$, should I present one solution, or the three solutions of $z^3=i$, $z\in\mathbb{C}$? If only one solution, what is the criteria? It seems principal roots can be ambiguous.

Answer 1

In this case I would give the three roots. When a single one needs to be picked the standard one is given by taking the more general definition of $z^w = \exp(w \log z)$ and take the principal value of $\log$.

Answer 2

Generally, for $z\in\mathbb C\setminus\{0\}$, there are $n$ different $n$-th roots $z^{1/n}$.

When $z\in\mathbb R^+$, we sometimes pick the unique root that is real-valued and positive, and ignore the others (depends on the context). But in your context, you would list all three of them, they are all equally valid.