(Easy 10 points!) Using DeMoivre's Theorem

Using DeMoivre's Theorem to evaluate the following. Express your answer in rectangular form.

a. (2-2isqr3)^6

b. The 4 fourth roots of -81.

Please show work! 10 points to best answer. Thanks!

a) The idea here is to do the pythagoras theorem between the two in the inside the brackets. Hence, we take out a magnitude (radius in the complex plane) of four. And our answer is 4^6 = 4096.

b) 3^4*(isinx + cosx)^4 = 81*(isin(4x) + cos(4x))

cos(4x) = -1 sin(4x) = 0

x = pi/4, 3*pi/4, 5*pi/4, 7*pi/4

the solutions make a square in the complex plane, with a magnitude of 3 radius, and we have:

(3/2)*(isqrt2 + sqrt2), (3/2)*(-isqrt2 + sqrt2), (3/2)*(isqrt2 - sqrt2), (3/2)*(-isqrt2 - sqrt2)