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.
