We have: #(- sqrt(3) - i)^(4)#
First, let's consider the complex number #z = - sqrt(3) - i#.
In order to apply De Moivre's theorem, we need to evaluate the modulus and argument of this #z#:
#=> |z| = sqrt((- sqrt(3)^(2) + (- 1)^(2))#
#=> |z| = sqrt(3 + 1)#
#=> |z| = sqrt(4)#
#=> |z| = 2#
#=> theta = arctan((- 1) / (- sqrt(3)))#
#=> theta = arctan((sqrt(3)) / (3))#
#=> theta = (pi) / (6)#
Then, #z# is located in the third quadrant:
#Rightarrow arg(z) = pi / 6 - pi = - (5 pi) / 6#
So, #z = 2 (cos(- (5 pi) / 6) + i sin(- (5 pi) / 6))#
Now, using De Moivre's theorem:
#=> z^(4) = 2^(4) (cos(4 cdot - (5 pi) / 6) + i sin(4 cdot - (5 pi) / 6))#
#=> z^(4) = 16 (cos(- (10 pi) / (3)) + i sin(- (10 pi) / (3)))#
#=> z^(4) = 16 (- (1) / (2) + (sqrt(3)) / (2) i)#
#=> z^(4) = 16 (- (1) / (2) (1 - sqrt(3) i))#
#=> z^(4) = - 8(1 - sqrt(3) i)#
#therefore z^(4) = - 8 + 8 sqrt(3) i#
Therefore, #(- sqrt(3) - i)^(4) = - 8 + 8 sqrt(3) i#.
