I want to find the value of $$i^{2/3}$$
Here was what I tried: $$i^{2/3} = (i^{2})^{1/3} = -1^{1/3} = (-1^{2})^{1/6} = 1$$
I know that I could have also stopped at the third step, since
$$-1^{1/3} = -1$$
Clearly there are multiple solutions, and I was wondering if there was a good way of conceptualizing these multiple solutions. How many solutions does a complex number to the $2/3$ power have, and can I use the complex plane to visualize this?
On
Alternatively, you could use the fact that $i = \cos (\pi/2)+ i \sin(\pi/2)$ to evaluate the power (https://en.wikipedia.org/wiki/Complex_number): $$i^{2/3} = \cos \left(\frac{\pi}{3} + \frac{4k\pi }{3}\right)+ i \sin\left(\frac{\pi}{3} + \frac{4k\pi}{3}\right), \\ \text{where}\ k = 0, 1, 2 $$
In complex analysis, $a^b$ is defined by $e^{b\log a}$, where the complex log, i.e., $\log a$, is multi-valued and given by $\log|a|+i (\arg(a)+2\pi k)$ ($k\in \mathbb{Z}$) (in $\log|a|$, we use the usual real log). In your case, we have $$i^{2/3}=e^{2/3\cdot(\log|i|+i(\arg i+2\pi k))} = e^{\pi i/3+4k\pi i/3}\,,$$ which takes three different values.