# Answer to Question #6069 in Complex Analysis for Jeannie

Question #6069

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

Expert's answer

(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)

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)

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