A Fifth Root

Algebra Level 3

Suppose \( z \) is a complex number such that \( z^5 = -\sqrt{3} + i\) and \( z = \sqrt[5]{2} ( \cos n^{\circ} + i \sin n^{\circ}) \), where \( 270 < n < 360 \). Find the exact value of \( n \).

Details and assumptions

\(i \) is the imaginary unit, where \(i^2=-1\).

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