10 The complex number \(z\), where \(0 < \arg z < \frac { 1 } { 2 } \pi\), is such that \(z ^ { 2 } = 3 + 4 \mathrm { i }\).
- Use an algebraic method to find \(z\).
- Show that \(z ^ { 3 } = 2 + 11 \mathrm { i }\).
The complex number \(w\) is the root of the equation
$$w ^ { 6 } - 4 w ^ { 3 } + 125 = 0$$
for which \(- \frac { 1 } { 2 } \pi < \arg w < 0\).
- Find \(w\).