Find the roots of complex number with only imaginary part

Question

I'm trying to solve the equation (find all the 5 roots) : $Z^5 = 32*i$, but with no success. Can anyone help me please?

Answer 1

Think geometrically. (Do you know the polar / exponential form of complex numbers?) How far away from the oprigin would such a root be? What angle would it make with the positive real axis?

Answer 2

So, $$\left(\dfrac z2\right)^5=i=e^{i\pi/2}=e^{i\pi(4n+1)/2}$$ where $n$ is any integer

Using De Moivre's formula and How to prove Euler's formula: $e^{it}=\cos t +i\sin t$? $$\dfrac z2=e^{i\pi(4n+1)/10}$$ where $n\equiv0,1,2,3,4\pmod5$