Question 3 - A-Level Maths

Exam Board	OCR
Module	Further Pure Core 1 (Further Pure Core 1)
Year	2023
Session	June
Topic	Complex numbers 2

Show that \(\frac { - 3 + \sqrt { 3 } \mathrm { i } } { 2 } = \sqrt { 3 } \mathrm { e } ^ { \frac { 5 } { 6 } \pi \mathrm { i } }\).
Hence determine the exact roots of the equation \(z ^ { 5 } = \frac { 9 ( - 3 + \sqrt { 3 } \mathrm { i } ) } { 2 }\), giving the roots in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\) where \(r > 0\) and \(0 \leqslant \theta < 2 \pi\).