Question 2 - A-Level Maths

Given that \(z _ { 1 } = 2 \mathrm { e } ^ { \frac { 1 } { 6 } \pi \mathrm { i } }\) and \(z _ { 2 } = 3 \mathrm { e } ^ { \frac { 1 } { 4 } \pi \mathrm { i } }\), express \(z _ { 1 } z _ { 2 }\) and \(\frac { z _ { 1 } } { z _ { 2 } }\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(0 \leqslant \theta < 2 \pi\).
Given that \(w = 2 \left( \cos \frac { 1 } { 8 } \pi + \mathrm { i } \sin \frac { 1 } { 8 } \pi \right)\), express \(w ^ { - 5 }\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\), where \(r > 0\) and \(0 \leqslant \theta < 2 \pi\).