Question and my solution below. Which part did I do wrong?

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Question and my solution below. Which part did I do wrong?

Mathematics
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At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.

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This is a review question and my book only has the solution & answers to odd number questions and this is an even number question. My answer is not even one of the choices lol
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I see no error in your working...

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are you sure? maybe the book is wrong. thanks for confirming it :) I'll probably ask my prof about that.
yup http://www.wolframalpha.com/input/?i=cube+roots+of+%28-3-3i%29
you are starting with \[\sqrt[3] { -3e^{\frac {\pi } {4} } } \]??
why is it -3? r is equal to \(\sqrt{18}\), so I have multiply it by the nth power which is 1/3 so it will be \(\sqrt[6]{18}\) since 1/2 times 1/3 is 1/6. De moivres + euler's formula: \(\large z^n=r^ne^{in \theta}\) sub the values in..it will be: \(\large z^{1/3}=\sqrt[6]{18}e^{i \frac{1}{3}( \frac{5 \pi}{4} + 2 \pi k)}\)
if we draw it first we have this |dw:1442695851885:dw| which kinda makes it easy as we have \(3 e^{i \frac{5 \pi}{4}}\)
ok. I got that.. but where did the "3" came from?
sorry we have \(\sqrt{18} e^{i \frac{5 \pi}{4}}\) which gives \(\large \sqrt[3]{\sqrt{18} \, e^{i(\frac{5 \pi}{4} + 2n\pi)} }\) \(\large = \sqrt[6]{18} \sqrt[3]{ \, e^{i(\frac{5 \pi}{4} + 2n\pi)} }\) \(\large =\sqrt[6]{18} \ \, e^{i(\frac{5 \pi}{12} + \frac{2n\pi}{3})} \) \(\large = \sqrt[6]{18} \ \, e^{i(\frac{5 \pi + 8n \pi}{12}) } \) \(\large \implies \sqrt[6]{18} \ \, e^{i(\frac{5}{12}) } \) \(\large \implies \sqrt[6]{18} \ \, e^{i(\frac{13}{12}) } \) \(\large \implies \sqrt[6]{18} \ \, e^{i(\frac{21}{12}) } \)
we got the same! :D the method is the same, I just used exponents instead of radical. Thanks for clarifying it also. I already sent a message to my prof. I'm just waiting for his response.
good luck!

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