3.9.8—Complex numbers

[r(cosθ+i sinθ)]^n=r^n(cos nθ+i sin nθ). To solve z^n=w, take the nth root of the modulus and add arguments (arg w+2kπ)/n for k=0,…,n−1.

List all distinct roots, then convert to rectangular form only if needed. Their Argand points lie equally spaced on a circle.

z³=8 has roots 2 cis(2kπ/3), k=0,1,2.

Taking only the principal argument yields one root and misses the remaining equally spaced solutions.