DeMoivre's Theorem?

Question

Can anyone help me to figure out the other answer. Thanks in advance
Find the two square roots of 4j by DeMoivre's Theorem. Give the roots in polar and rectangular form.

I think one answer is 4 angle 90 degree

Answer 1

I think you mean 4i, not 4j :P

On a diagram, 4i is directly up at a distance of 4, ie 4 cis 90.
So, (4 cis 90)^0.5 = (4^0.5) cis (90 * 0.5) = 2 cis 45.
For the other one, add (or subtract) 360 * 0.5 = 180 to the angle to get 2 cis 225, or 2 cis -135.

Those becomes sqrt(2) + i*sqrt(2) and -sqrt(2) - i*sqrt(2).

Answer 2

do you mean 4i??
i'm going to assume you do

4i = 4 * e^(pi/2 * i +2pi *k) for k an integer

sqrt(4i) = 2 * e^(pi/4 i + pi * k)
= 2 * e^(pi/4) and 2 * e^(5pi/4) (note, if you add any more copies of pi, you get back one of those two answers)

so in polar form, your answers are (2, pi/4) and (2, 5pi/4)

using demoivres theorem, in rectangular coordinates:
(2, pi/4) = 2(cos (pi/4) + i sin(pi/4)) = 2(sqrt(2)/2 + i * (sqrt(2)/2))
= sqrt(2) + sqrt(2) * i

(2, 5pi/4) = - sqrt(2) - sqrt(2) * i