Roots of unity and special equations

A question is this sub-type if and only if it involves finding roots of equations of the form z^n = c (including roots of unity) or other special polynomial forms that can be solved using exponential/polar form.

1 questions

Edexcel CP2 2024 June Q7
  1. (a) Determine the roots of the equation
$$z ^ { 6 } = 1$$ giving your answers in the form \(\mathrm { e } ^ { \mathrm { i } \theta }\) where \(0 \leqslant \theta < 2 \pi\)
(b) Show the roots of the equation in part (a) on a single Argand diagram.
(c) Show that $$( \sqrt { 3 } + i ) ^ { 6 } = - 64$$ (d) Hence, or otherwise, solve the equation $$z ^ { 6 } + 64 = 0$$ giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\) where \(0 \leqslant \theta < 2 \pi\)