 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2015/paper-2/2/solution

Write

- 8 = 8 e^{i (π + 2 πk)}

. Its three roots have modulus

2

and arguments

(π + 2 πk) /3

, with

k = 0, 1, 2

. Thus all three complex roots are

z = 1 + i 3, - 2, 1 - i 3 .

Further integer values of

k

repeat these roots.

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