Roots of Complex Equation

Question

Find the 3rd Root of equation $z= 8\left(\cos(\frac \pi 4) + i \sin(\frac \pi 4)\right)$

If I write the values of cos and sin then I would have only one root, how to find 3 roots for above equation ?

Answer 1

This equation has only one solution for $z$ (the given one). This one can also be written as $z = 8 e^{i\pi/4}$.

Perhaps you might want to find the 3rd root of $z$?

Then youre gonna have the solutions

$$"\sqrt[3]{z}" = \sqrt[3]{8} e^{i\pi/12} e^{i2 \pi k/ 3} = 2 e^{i2\pi(1/24+k/3)}$$ for $k = 0,1,2$