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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
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?
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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.
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!