 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2014/paper-1/12/a/solution

- 1 + i

2

and argument

3 π /4

(2)^{1/3} = 2^{1/6}

, and its argument is one third of

3 π /4 + 2 kπ

. Hence the three distinct roots are

z_{k} = 2^{1/6} exp [i (\frac{π}{4} + \frac{2 kπ}{3})], k = 0, 1, 2.

Their displayed arguments are

π /4

11 π /12

, and

19 π /12

; the last is equivalently

- 5 π /12

in the principal range.

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