2 You are given the complex numbers \(z = \sqrt { 32 } ( 1 + \mathrm { j } )\) and \(w = 8 \left( \cos \frac { 7 } { 12 } \pi + \mathrm { j } \sin \frac { 7 } { 12 } \pi \right)\).

1. Find the modulus and argument of each of the complex numbers \(z , z ^ { * } , z w\) and \(\frac { z } { w }\).
2. Express \(\frac { z } { w }\) in the form \(a + b \mathrm { j }\), giving the exact values of \(a\) and \(b\).
3. Find the cube roots of \(z\), in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
4. Show that the cube roots of \(z\) can be written as $$k _ { 1 } w ^ { * } , \quad k _ { 2 } z ^ { * } \quad \text { and } \quad k _ { 3 } \mathrm { j } w ,$$ where \(k _ { 1 } , k _ { 2 }\) and \(k _ { 3 }\) are real numbers. State the values of \(k _ { 1 } , k _ { 2 }\) and \(k _ { 3 }\).