Roots of complex numbers and roots of unityEdexcel A-Level Further Maths: Mind map

Exam questions on Roots of complex numbers and roots of unity

1. The complex number $w=8\mathrm{i}$ and the equation $z^3=8\mathrm{i}$.
   Find the three roots of $z^3=8\mathrm{i}$, giving each in the form $x+\mathrm{i}y$ where $x$ and $y$ are exact.2 marks
2. Let $\omega=\mathrm{e}^{2\pi\mathrm{i}/5}$, so that $1,\omega,\omega^2,\omega^3,\omega^4$ are the fifth roots of unity.
   Show that $\omega^4=\overline{\omega}$, and hence show that $\omega+\omega^4=2\cos\frac{2\pi}{5}$.2 marks
3. The complex number $w=-4+4\sqrt3\,\mathrm{i}$.
   Write $w$ in the form $r\mathrm{e}^{\mathrm{i}\theta}$, where $r>0$ and $-\pi<\theta\le\pi$.3 marks