Multiply angles and find every root

In polar form, multiplication multiplies moduli and adds arguments. De Moivre’s theorem extends this to integer powers. A negative power takes the reciprocal modulus and reverses the angle multiplier.

An nth root has modulus equal to the nth root of the original modulus. Include all angles differing by full turns before dividing by n: this produces n equally spaced roots.

\displaystyle (r\operatorname{cis}\theta)^n=r^n\operatorname{cis}(n\theta),\quad z_k=r^{1/n}\operatorname{cis}\frac{\theta+2k\pi}{n}

EXPLORE THE ARGAND PLANEReal part: 3Imaginary part: 2Modulus 3.61 · Argument 33.7°

WORKED EXAMPLE

Find all cube roots of 8.

The root modulus is 2.
Use arguments (0 + 2kπ)/3 for k = 0, 1, 2.
The roots are 2, −1 + $\sqrt{3}$ i and −1 − $\sqrt{3}$ i.

Assumed knowledge

Use complex arithmetic, modulus and argument, and convert between Cartesian and polar form. Recall radians, coterminal angles, index laws and exact trigonometric values.

Study Specialist Mathematics alongside, or after, Mathematical Methods. Later units build on earlier units.

QCAA prior-learning guidance ↗

Multiply angles and find every root

Find all cube roots of 8.

Put the idea to work.