11 In this question you must show detailed reasoning.
In Fig. 11, the points \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E }\) and F represent the complex sixth roots of 64 on an Argand diagram. The midpoints of \(\mathrm { AB } , \mathrm { BC } , \mathrm { CD } , \mathrm { DE } , \mathrm { EF }\) and FA are \(\mathrm { G } , \mathrm { H } , \mathrm { I } , \mathrm { J } , \mathrm { K }\) and L respectively.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c2be8838-50ec-4e82-b203-4608ab56c110-5_807_872_443_239}
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\caption{Fig. 11}
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- Write down, in exponential ( \(r \mathrm { e } ^ { \mathrm { i } \theta }\) ) form, the complex numbers represented by the points \(\mathrm { A } , \mathrm { B }\), \(\mathrm { C } , \mathrm { D } , \mathrm { E }\) and F .
- When these complex numbers are multiplied by the complex number \(w\), the resulting complex numbers are represented by the points G, H, I, J, K and L.
Find \(w\) in exponential form.
- You are given that \(\mathrm { G } , \mathrm { H } , \mathrm { I } , \mathrm { J } , \mathrm { K }\) and L represent roots of the equation \(z ^ { 6 } = p\).
Find \(p\).