Use DeMoivres Theorem and leave the answer in a bi form Squ

Use DeMoivre\'s Theorem and leave the answer in a + bi form. (-Squareroot 3 + i)^6

Solution

( - sqrt 3 + i )^6

z = r ( cos theta + i sin theta )

r = |z| = sqrt (a^2 + b^2 )

r = sqrt ( 3 + 1) = 2

x = r cos theta , y = r sin theta

cos theta = - sqrt 3 / 2 , sin theta = 1 / 2

theta = 5pi/6

( - sqrt 3 + i )^6 = [ 2 ( cos 5pi/6 + i sin 5pi/6 )] ^6

= 2^6 ( cos 6*5pi/6 + i sin 5*6 pi/6 ]

= 64 ( cos 30pi/6 + i sin 30 pi/6 )

= 64 ( -1 + 0 i )

= -64 + 0i