I don't well understand how WA performs the exponentials in the complex field.
I see that a simple case as $(-1)^{2/3}$ is treated as $((-1)^{1/3})^2$ so it gives (for the principal values): $$ (-1)^{2/3}=((-1)^{1/3})^2=(e^{i\pi/3})^2=e^{i(2\pi/3)}=-\frac{1}{2}+i\frac{\sqrt{3}}{2} $$
that is different from $$ ((-1)^2)^{1/3}=1 $$
In the same way, for $(-1)^{4/3}$ we find:
$$
(-1)^{4/3}=e^{i4\pi/3}=-e^{i\pi/3}=-\frac{1}{2}-i\frac{\sqrt{3}}{2}
$$
Now, as I aspect, for the equation:
$ z^{3/2}=-1$ WA gives the result $-\frac{1}{2}+i\frac{\sqrt{3}}{2}$, but, for $ z^{3/4}=-1$ WA says that there are no solutions.
So it seems to me that these answers of WA are not coherent, but maybe that I'm wrong and simply I don't understand some subtle problem.
This question come from my comment to: Do $z^{3/4}=-1 $ solutions exist?