(i) Write down the five fifth roots of unity.
(ii) Hence find all the roots of the equation
\(z^{5}+16+(16 \sqrt{ } 3) i=0\)
giving answers in the form \(r \mathrm{e}^{\mathrm{i} q \pi}\), where \(r\gt 0\) and \(q\) is a rational number. Show these roots on an Argand diagram.

Let \(w\) be a root of the equation in part (ii).
(iii) Show that
\(\sum_{k=0}^{4}\left(\frac{w}{2}\right)^{k}=\frac{3+i \sqrt{ } 3}{2-w} .\)
(iv) Identify the root for which \(|2-w|\) is least.