How is the principle root defined?

Question

I always thought that the principle root of $n$ was defined as the only positive real solution for $n$. But what is defined as the principal root for numbers with no positive real $n$th roots? An example would be $\sqrt[3]{-1}$. Wolfram Alpha says its principle root is $\frac{1+i*\sqrt[2]{3}}{2}$, but why not its conjugate? And if it's defined as the root $a$ that fits the criteria Rez(a)>0, what about numbers that have more than one root that fit this criteria? Thanks in advance :]