Complex Roots - Not sure I did this right

[Atena]

Hello. I'm not sure whether I did this right or messed up somewhere, just need to confirm my results...thanks to anybody who bothers answering.

Homework Statement

Find all the roots of z^{4}=1-i

Homework Equations

I guess I should state De Moivre's here...

(r cis(\vartheta))^{n}=r^{n} cis (n\vartheta)

The Attempt at a Solution

Firstly I re-wrote z^{4}=1-i as

z^{4}=\sqrt{2} cis (\frac{-\pi}{4})

Using De Moivre's,

z=(2\frac{1}{2})^{\frac{1}{4}} cis (\frac{1}{4}(\frac{-\pi}{4}+k2\pi))

z=2\frac{1}{8} cis (\frac{1}{4}(\frac{-\pi}{4}+k2\pi))

I found the four roots letting k=0,1,2,3

z=2^\frac{1}{8} cis (\frac{-\pi}{16})

z=2^\frac{1}{8} cis (\frac{1}{4}(\frac{-\pi}{4}+2\pi))=2^\frac{1}{8} cis (\frac{7\pi}{16})

z=2^\frac{1}{8} cis (\frac{1}{4}(\frac{-\pi}{4}+4\pi))=2^\frac{1}{8} cis (\frac{15\pi}{16})

z=2^\frac{1}{8} cis (\frac{1}{4}(\frac{-\pi}{4}+6\pi))=2^\frac{1}{8} cis (\frac{23\pi}{16})

 

[yyat]

Hi Atena!

Your answer is correct. You can check this by taking the fourth powers of the solutions you got (using deMoivre).