N1993 P2 Q12

Write down the modulus and argument of the complex number \(w\), where \(w=1+\mathbf{i}\sqrt{3}\). Hence find the set of values of the positive integer \(n\) for which \({{w}^{n}}\)is real, and show that, for these values of \(n\), the value of \({{w}^{n}}\) is either \({{2}^{n}}\)or \(-{{2}^{n}}\).

Express \({{w}^{100}}-{{\left( w^* \right)}^{100}}\) in the form \(k\mathbf{i}\), where \(k\) is real, giving the exact value of \(k\) in non-trigonometrical form.