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.
9 (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 $r > 0$ 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
$$\sum _ { k = 0 } ^ { 4 } \left( \frac { w } { 2 } \right) ^ { k } = \frac { 3 + \mathrm { i } \sqrt { } 3 } { 2 - w }$$
(iv) Identify the root for which $| 2 - w |$ is least.
\hfill \mbox{\textit{CAIE FP1 2010 Q9}}