State De Moivre's theorem for a positive integer \(n\).

Complete the formula: if \(z=r\operatorname{cis}\theta\), then \(z^n=\ ?\).

Use De Moivre's theorem to simplify \((\operatorname{cis}(\pi/5))^3\).

Find \((2\operatorname{cis}(\pi/6))^2\) in polar form.

Use De Moivre's theorem to find \((1+i)^4\).

Use De Moivre's theorem to find \((\sqrt3+i)^3\).

Find \((-1+i)^6\) using polar form.

Find \((2-2i)^3\) using De Moivre's theorem.

Find all square roots of \(4\operatorname{cis}(\pi/3)\).

Find the cube roots of \(8\).

Explain why the \(n\)-th roots of a non-zero complex number are evenly spaced around a circle.

A student finds only one cube root of \(-8\), namely \(-2\). Explain the missing roots.

Use De Moivre's theorem to express \(\cos 3\theta\) in terms of \(\cos\theta\).

Use De Moivre's theorem to express \(\sin 3\theta\) in terms of \(\sin\theta\).

Find all fourth roots of \(16\operatorname{cis}(2\pi/3)\).

Find the possible values of \(n\in\mathbb N\) with \(1\le n\le 8\) for which \((\operatorname{cis}(\pi/4))^n=-1\).

For which integers \(n\) with \(1\le n\le 12\) is \((1+i)^n\) real?

Prove De Moivre's theorem for positive integers using exponential form.

Show that the product of all three cube roots of \(8\) is \(8\).

A student uses only the principal argument when finding roots and gets one root instead of \(n\). Explain why adding \(2k\pi\) before division is necessary.