Express in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\) :
\(\quad 4 ( 1 + i \sqrt { 3 } )\);
\(4 ( 1 - i \sqrt { 3 } )\).
The complex number \(z\) satisfies the equation
$$\left( z ^ { 3 } - 4 \right) ^ { 2 } = - 48$$
Show that \(z ^ { 3 } = 4 \pm 4 \sqrt { 3 } \mathrm { i }\).
Solve the equation
$$\left( z ^ { 3 } - 4 \right) ^ { 2 } = - 48$$
giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
Illustrate the roots on an Argand diagram.
Explain why the sum of the roots of the equation
$$\left( z ^ { 3 } - 4 \right) ^ { 2 } = - 48$$
is zero.