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

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ZeHanz
 3 years ago
Best ResponseYou've already chosen the best response.1If you draw the number in the complex plane, (draw point at coordinates (1, sqrt(3))), you get a wellknown triangle: 306090 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...)

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0It sounds extremely difficult. Isn't there another way to do this without using the plane?

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0Nevermind. I figured it out! Thank you so much.

ZeHanz
 3 years ago
Best ResponseYou've already chosen the best response.1YOu know, calculating (a+bi)^10 by working out the brackets is unbelievably more difficult! You would have to expand (1sqrt(3))(1sqrt(3))......(1sqrt(3)) :( Now the other way. See drawing. z=1sqrt(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 1sqrt(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 1sqrt(3) to: 512512sqrt(3) and that is the final answer!
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