Fourth roots of -16

Problem

Find the fourth roots of $-16$ and plot them in the Argand-Gauss plane.

Solution

Write $-16 = 16\,e^{\,i\pi}$, so $\rho=16$ and $\sqrt[4]{16}=2$. The fourth roots are $w_k = 2\,\exp\!\left(i\,\frac{\pi + 2k\pi}{4}\right), \qquad k=0,1,2,3:$ $\begin{aligned} w_0 &= 2\,e^{\,i\pi/4} = \sqrt 2 + i\sqrt 2, & w_1 &= 2\,e^{\,i3\pi/4} = -\sqrt 2 + i\sqrt 2, \\ w_2 &= 2\,e^{\,i5\pi/4} = -\sqrt 2 - i\sqrt 2, & w_3 &= 2\,e^{\,i7\pi/4} = \sqrt 2 - i\sqrt 2. \end{aligned}$ The four roots are the vertices of a square inscribed in the circle of radius $2$.

The four fourth roots of $-16$ as the vertices of a square inscribed in the circle of radius $2$.

Links

Topics: Complex numbers
Concepts: Exponential form · Argand-Gauss plane · nth roots
Methods: Complex numbers in polar form
Skills: Calculating · Using formulae
Exercise type: Evaluating expressions