# Answer to Question #2827 in Complex Analysis for alyssa

Question #2827

Find all third roots of -64i .

Expert's answer

Denote x =

-64i= 64 (cos (3π/2)+i sin (3π/2))

Hence the set of root of -64i of degree 3 consists of the following three numbers:

X1 = 4 (cos (3π/(2∙3)) + i sin (3π/(2∙3)))= 4 (cos π/2+i sin π/2) = 4i

X2 = 4 (cos (7π/(2∙3)) + i sin (7π/(2∙3)))= 4 (-√3/2-i 1/2) = -2√3-2i

X3 = 4 (cos(11π/(2∙3)) + i sin (11π/(2∙3)))=& 4 (√3/2-i 1/2) = 2√3-2i

-64i= 64 (cos (3π/2)+i sin (3π/2))

Hence the set of root of -64i of degree 3 consists of the following three numbers:

X1 = 4 (cos (3π/(2∙3)) + i sin (3π/(2∙3)))= 4 (cos π/2+i sin π/2) = 4i

X2 = 4 (cos (7π/(2∙3)) + i sin (7π/(2∙3)))= 4 (-√3/2-i 1/2) = -2√3-2i

X3 = 4 (cos(11π/(2∙3)) + i sin (11π/(2∙3)))=& 4 (√3/2-i 1/2) = 2√3-2i

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