Find the modulus of the complex number \(- 4 \sqrt { 3 } + 4 \mathrm { i }\), giving your answer as an integer.
The locus of points, \(L\), satisfies the equation \(| z + 4 \sqrt { 3 } - 4 \mathrm { i } | = 4\).
Sketch the locus \(L\) on the Argand diagram below.
The complex number \(w\) lies on \(L\) so that \(- \pi < \arg w \leqslant \pi\).
Find the least possible value of arg \(w\), giving your answer in terms of \(\pi\).
Solve the equation \(z ^ { 3 } = - 4 \sqrt { 3 } + 4 \mathrm { i }\), giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\). [0pt]
[5 marks]