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For $z\ne 0$, the principal value of complex exponentiation is $z^w=\exp (w\mathrm{Log}z)$, where the principal complex logarithm is $\mathrm{Log}z=\log |z|+i\mathrm{Arg}z$ with $-\pi <\mathrm{Arg}z\le \pi$. De Moivre's theorem states that $(\cos \theta +i\sin \theta {)}^n=\cos (n\theta )+i\sin (n\theta )$ for every integer $n$.

Since $\sqrt{3}+i=2e^{i\pi /6}$, the six roots are

Use the principal complex logarithm, for which $\mathrm{Log}i=i\pi /2$, and setThen the definition of complex exponentiation givesand thereforeOther choices of branch of the complex logarithm produce further valid answers.

By complex exponentiation, a value of $z^i$ has the formFor this to equal one, its exponent must be $2\pi in$ for some $n\in \mathbb{Z}$. Hence a logarithm of $z$ must equal $2\pi n$, so $z$ is positive real andWith the principal branch this is the same list; in the multivalued convention these are precisely the $z$ for which one value of $z^i$ equals one.

The logarithms of $1+\omega =e^{i\pi /3}$ areThe definition of complex exponentiation therefore gives all values asSince $1+\omega =\frac{1}{2}+i\frac{\sqrt{3}}{2}$, these can be written