cube root of i without refrence to i

Question

Is it possible to express the cube root of "i" without using "i" itself?

If this is possible can you show me how to arrive at it?

thanks

Answer 1

On the unit circle mark the $30$ degree, $150 $ degree and $270$ degree points. These are the cube roots of $i$

Answer 2

No. $i$ has 3 cube roots in the complex numbers and none of them can of course be real (i.e. have no imaginary part; a real number has a real third power, not $i$). They are

$$e^{i\frac{\pi}{6}}, e^{i\frac{5\pi}{6}}, e^{i\frac{3\pi}{2}} $$

which you can write out using Euler's $e^{it} = \cos(t) + i\sin(t)$ as usual.