Find (–1 –i sqrt(3))^10 .Express in rectangular form.

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Find (–1 –i sqrt(3))^10 .Express in rectangular form.

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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If you draw the number in the complex plane, (draw point at coordinates (-1, -sqrt(3))), you get a well-known triangle: 30-60-90 degrees, and you can easily find the hypothenuse. This means, if z = -1 -sqrt(3), you now have |z| and arg(z). Now remember: if you calculate w = z^10, then |w|=|z|^10 and arg(w) = 10arg(z). Then you could draw the new number w also on the complex plane. Again, you will get a nice triangle, so you can write w in the form a+bi without any problem. (Sounds more difficult than it is...)
It sounds extremely difficult. Isn't there another way to do this without using the plane?
Nevermind. I figured it out! Thank you so much.

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YOu know, calculating (a+bi)^10 by working out the brackets is unbelievably more difficult! You would have to expand (-1-sqrt(3))(-1-sqrt(3))......(-1-sqrt(3)) :( Now the other way. See drawing. z=-1-sqrt(3) means: |z| =2. You have to calculate z^10 This means |z^10| = 2^10=1024. So the number you are looking for has modulus (absolute value, magnitude) 1024. We're already halfway now! The argument of -1-sqrt(3) is 240 degrees. This means the argument of the 10th power is 10 * 240 = 2400 degrees. 2400 degrees = 2400/360=6 2/3 circles, or 2/3 circle, or 240 degrees! So the number you are looking for, lies exactly in the same direction as the original one, just a "bit" further out: at 1024 instead of 2 from 0. That is 512 times further away... this means it has gone from -1-sqrt(3) to: -512-512sqrt(3) and that is the final answer!
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Thank you so much
yw!

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