Complex Arithmetic - Mathematica agrees with me, textbook says I'm wrong.

[jdinatale]

Mathematica agrees with my second solution (not the first one though). The back of my textbook says: "$\sqrt[4]{8}[\cos(\frac{5\pi}{8}) + i\sin(\frac{5\pi}{8})]$ and $\sqrt[4]{8}[\cos(\frac{13\pi}{8}) + i\sin(\frac{13\pi}{8})]$"


Edit: The second z in my picture should be |z|, the modulus.

Edit2: Here are the results from mathematica, confirming that both the textbook AND I are right...

http://www.wolframalpha.com/input/?i=(1+-+i)^(3/2)+=+(8^(1/4))[cos(45pi/8)+++i+sin(45pi/8)]
http://www.wolframalpha.com/input/?i=(1+-+i)^(3/2)+=+[8^(1/4)][cos(+13pi/8)+++isin(13pi/8)]

My question is, how did my textbook get that answer?

 

[vela]

You can get rid of multiples of $2\pi$. For example, you have
$$\frac{21\pi}{8} = \frac{(16+5)\pi}{8} = 2\pi + \frac{5\pi}{8},$$ so your first answer is equivalent to the book's first answer. Similarly, you can show your second answer also matches the book's second answer.