(a) Find the roots of the equation \(z^{3}=-1-\mathrm{i}\), giving your answers in the form \(r \mathrm{e}^{\mathrm{i} \theta}\), where \(r>0\) and \(0 \leqslant \theta<2 \pi\).

Let \(w=z_{1}^{3 k}+z_{2}^{3 k}+z_{3}^{3 k}\), where \(k\) is a positive integer and \(z_{1}, z_{2}, z_{3}\) are the roots of \(z^{3}=-1-\mathrm{i}\).
(b) Express \(w\) in the form \(R \mathrm{e}^{\mathrm{i} \alpha}\), where \(R>0\), giving \(R\) and \(\alpha\) in terms of \(k\).