Answer:
(2, 4π/15), (2, 14π/15), (2, 24π/15)
Step-by-step explanation:
DeMoivre's theorem tells you the n-th root of a complex number in polar form is ...
(magnitude, angle)^(1/n) = (magnitude^(1/n), (angle +2kπ)/n) for k = 0 to n-1.
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Your number has a magnitude of 8, so the cube root of that is 2.
Your number has an angle of (4π/5+2kπ), so one third of that is ...
(π/3)(4/5 +2k) . . . for k = 0, 1, 2
Then the cube roots are (magnitude, angle) ...
{(2, 4π/15), (2, 14π/15), (2, 24π/15)}
Of course, you can write (magnitude, angle) in CIS form as ...
magnitude(cos(angle) +i·sin(angle))
as may be required by your grader.
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Comment on complex number notation
The notation used in my engineering courses was fairly practical. A complex number could be written as a+bi or as magnitude∠angle. We didn't waste effort writing it as magnitude(cos(angle) +i·sin(angle)) and we avoided the confusion associated with different interpretations of an ordered pair.
