Can anyone help me with finding the solutions to:

(z^8)-256=0

I can find z= 2, -2, 2i, -2i but don't know how to find the other 4 solutions, apparently I can let z^2=(a+ib)^2 and solve for a and b, but don't know how to do this. Thanks in advance.

There are n different nth roots of z=r(cos{\theta}+i\cdot sin{\theta})

Therefore, these are given by:

\sqrt[n]{r}\left[cos(\frac{\theta}{n}+\frac{2k{\pi}}{n})+i\cdot sin(\frac{\theta}{n}+\frac{2k{\pi}}{n})\right]

Where \theta=2{\pi} and n=8, so we have \frac{2\pi}{8}=\frac{\pi}{4}

\sqrt[8]{256}\left[cos(\frac{\pi}{4}+\frac{2k{\pi}}{8})+i\cdot sin(\frac{\pi}{4}+\frac{2k{\pi}}{8})\right]

and let k=1,2,3...

If k=1, we get 2i

If k=2, we get -\sqrt{2}+\sqrt{2}i

and so on.

Hey thanks I get confused about which form to express it in to work out the solution, I had tried to put it in polar exponential form and didn't think on expanding etc.


There are n different nth roots of z=r(cos{\theta}+i\cdot sin{\theta})

Therefore, these are given by:

\sqrt[n]{r}\left[cos(\frac{\theta}{n}+\frac{2k{\pi}}{n})+i\cdot sin(\frac{\theta}{n}+\frac{2k{\pi}}{n})\right]

Where \theta=2{\pi} and n=8, so we have \frac{2\pi}{8}=\frac{\pi}{4}

\sqrt[8]{256}\left[cos(\frac{\pi}{4}+\frac{2k{\pi}}{8})+i\cdot sin(\frac{\pi}{4}+\frac{2k{\pi}}{8})\right]

and let k=1,2,3.....

If k=1, we get 2i

If k=2, we get -\sqrt{2}+\sqrt{2}i

and so on.