We have: (- sqrt(3) - i)^(4)(−√3−i)4
First, let's consider the complex number z = - sqrt(3) - iz=−√3−i.
In order to apply De Moivre's theorem, we need to evaluate the modulus and argument of this zz:
=> |z| = sqrt((- sqrt(3)^(2) + (- 1)^(2))⇒|z|=√(−√32+(−1)2)
=> |z| = sqrt(3 + 1)⇒|z|=√3+1
=> |z| = sqrt(4)⇒|z|=√4
=> |z| = 2⇒|z|=2
=> theta = arctan((- 1) / (- sqrt(3)))⇒θ=arctan(−1−√3)
=> theta = arctan((sqrt(3)) / (3))⇒θ=arctan(√33)
=> theta = (pi) / (6)⇒θ=π6
Then, zz is located in the third quadrant:
Rightarrow arg(z) = pi / 6 - pi = - (5 pi) / 6⇒arg(z)=π6−π=−5π6
So, z = 2 (cos(- (5 pi) / 6) + i sin(- (5 pi) / 6))z=2(cos(−5π6)+isin(−5π6))
Now, using De Moivre's theorem:
=> z^(4) = 2^(4) (cos(4 cdot - (5 pi) / 6) + i sin(4 cdot - (5 pi) / 6))⇒z4=24(cos(4⋅−5π6)+isin(4⋅−5π6))
=> z^(4) = 16 (cos(- (10 pi) / (3)) + i sin(- (10 pi) / (3)))⇒z4=16(cos(−10π3)+isin(−10π3))
=> z^(4) = 16 (- (1) / (2) + (sqrt(3)) / (2) i)⇒z4=16(−12+√32i)
=> z^(4) = 16 (- (1) / (2) (1 - sqrt(3) i))⇒z4=16(−12(1−√3i))
=> z^(4) = - 8(1 - sqrt(3) i)⇒z4=−8(1−√3i)
therefore z^(4) = - 8 + 8 sqrt(3) i
Therefore, (- sqrt(3) - i)^(4) = - 8 + 8 sqrt(3) i.
