4. (a) Given that $$z = r ( \cos \theta + \mathrm { i } \sin \theta ) , \quad r \in \mathbb { R }$$ prove, by induction, that \(z ^ { n } = r ^ { n } ( \cos n \theta + \mathrm { i } \sin n \theta ) , \quad n \in \mathbb { Z } ^ { + }\) $$w = 3 \left( \cos \frac { 3 \pi } { 4 } + i \sin \frac { 3 \pi } { 4 } \right)$$ (b) Find the exact value of \(w ^ { 5 }\), giving your answer in the form \(a + \mathrm { i } b\), where \(a , b \in \mathbb { R }\).