Find the nth roots of a complex number and the nth roots of unity, and represent them on the Argand plane

What this dot point is asking

SCSA wants you to solve $z^n = w$ for all roots, identify the roots of unity, and plot them as a regular polygon on the Argand plane.

The general root formula

To solve $z^n = w$, write $w = R\,\text{cis}\,\phi$ and let $z = \rho\,\text{cis}\,\alpha$. By de Moivre $z^n = \rho^n\,\text{cis}(n\alpha)$, so matching moduli gives $\rho = R^{1/n}$ and matching arguments gives $n\alpha = \phi + 2\pi k$ for integer $k$. Hence

Taking $k = 0$ to $n-1$ gives $n$ distinct roots; further values of $k$ repeat them. All roots share the modulus $R^{1/n}$, so they lie on a circle of that radius, spaced $\tfrac{2\pi}{n}$ apart. This is why they form a regular $n$-gon.

The nth roots of unity

Setting $w = 1 = \text{cis}\,0$ gives the $n$th roots of unity:

They all lie on the unit circle, include $z = 1$, and are equally spaced. Writing $\omega = \text{cis}\tfrac{2\pi}{n}$, every root is a power $\omega^k$.

Plotting strategy

Find one root (usually $k = 0$), then rotate by $\tfrac{2\pi}{n}$ repeatedly to get the rest. You never recompute the modulus.