2024 MI PRU2 P1 Q7

It is given that \(1+a\,\mathbf{i}\) is one of the roots of the equation \({{w}^{3}}+8=0\), where \(a\) is a positive constant. Without using a calculator, find \(a\) and the other roots. [4]

Deduce the roots of \({{w}^{4}}+{{w}^{3}}+8w+8=0\). [2]

Hence determine the roots of the equation \({{z}^{4}}-\mathbf{i}\,{{z}^{3}}+8\mathbf{i}\,z+8=0\). [2]