Say that for some complex number $w$

$$e^{wi} = a$$

Now raise both sides to $1/4$.

$$e^{wi/4} = a^{1/4}$$

Now $e^{wi/4}$ has a single defined value. Yet $a^{1/4}$ can have multiple values. So why is Euler's formula well defined for non-integer powers, since non-integer powers can yield multiple values?