Answer to Question #6069 in Complex Analysis for Jeannie

Question

Find the following complex numbers in the form x + iy. a) (1-(sqrt3)i)^85

Answer 1

(1-(sqrt3)i)=2(1/2-(sqrt3)i/2)=2(cos(-pi/3)+i sin(-pi/3))
Now we have our complex number in trigonometric form
Then we use de Moivre's formula which states that&
Therefore [2(cos(-pi/3)+i sin(-pi/3))]^85=2^85*(cos(-85*pi/3)+i sin(-85*pi/3))

85*pi/3=24pi+pi/3=12*2pi+pi/3
Hence 2^85*(cos(-85*pi/3)+i sin(-85*pi/3))=2^85(cos(-12*2pi-pi/3)+i sin(-12*2pi-pi/3))=
=2^85*(cos(-pi/3)+i sin(-pi/3))=2^85*(1/2-i (sqrt3)/2)=2^84*(1-i(sqrt3))=2^84*-i *2^84*(sqrt3)
Answer: 2^84*-i *2^84* (sqrt3)