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 \(r > 0\) and \(q\) is a rational number. Show these roots on an Argand diagram.
Let \(w\) be a root of the equation in part (ii).
Show that
$$\sum _ { k = 0 } ^ { 4 } \left( \frac { w } { 2 } \right) ^ { k } = \frac { 3 + \mathrm { i } \sqrt { } 3 } { 2 - w }$$
Identify the root for which \(| 2 - w |\) is least.