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

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?