Exam Questions - nth roots of a complex number

nth roots of a complex number

Roots of numbers are not unique. A root of a number x is y such that when y is raised to the power of n, the result is x. Every non-zero complex number has n nth roots which are evenly dispatched around a circle in the complex plane.
The nth roots of a complex number z = r(cos θ + i sin θ), are given by z^(1/n) = r^(1/n) [cos ((θ + 2kπ)/n) + i sin ((θ + 2kπ)/n)] where k= 0,1,2,3…n-1.
Each root corresponds to a different value of k.

To find the nth roots of a complex number, first express the complex number in polar form, i.e., as a magnitude r and an angle θ.
Once you have it in polar form, find the nth root of the modulus, r^(1/n).
Divide the original angle θ by n to calculate the angle of each of the nth roots.
Determine all roots by adding multiples of 2π/n to the angle. Each multiple of 2π/n corresponds to a different root.
You should have n different roots. Each root is spaced at equal angles apart in the complex plane.

The nth roots of a complex number can be visualistically represented set of points on a circle in the complex plane, equally spaced around the origin.
The modulus of the roots is the nth root of the modulus of the original complex number and the argument of each root are equally spaced and divide the circle (of 360 degrees or 2π radians) into n equal parts.

De Moivre’s Theorem is highly applicable when dealing with nth roots of complex numbers. It states that (cos θ + i sin θ)^n = cos nθ + i sin nθ, and can be derived from the multiplication and division rules for complex numbers.

Always express complex numbers in polar form when finding nth roots.
Different roots correspond to different multiples of 2π/n added to the original angle.
All nth roots will have the same modulus and will be spaced equally apart on the complex plane.
De Moivre’s Theorem is a helpful tool for finding and manipulating complex number roots.
Knowing these properties of roots and being able to apply them to solve problems is a critical skill when dealing with complex numbers in further mathematics.