nth Roots of a Complex Number

Introduction to nth Roots of Complex Numbers

Complex numbers are not only limited to square roots or cube roots, but they can have nth roots as well.
If you have a complex number a + bi, the nth root of this complex number is another complex number which when raised to the power of n gives the original number.
There will always be n different nth roots for any given complex number in the complex plane.
For example, a complex number will have 5 fifth roots, 7 seventh roots and so on.

To calculate nth roots, the complex number is usually written in its polar form: r(cos θ + i sin θ), where r is magnitude and θ is the argument (in radians).
Using De Moivre’s Theorem, the nth roots are given by: r^(1/n)[cos ((θ + 2kπ) / n) + i sin((θ + 2kπ) / n)], where k ranges from 0 to n - 1.
Each distinct value of k will give a different nth root.

Compute the magnitude and argument of the complex number first (remember to convert the argument to radians if necessary).
Use De Moivre’s Theorem as mentioned above to find the nth roots.
Don’t forget, for each root, increment k by 1 in the De Moivre’s equation until you reach n - 1.

Want to find the cube roots of 8 (which can be written as 8(cos 0 + i sin 0) in polar form).
Here, r = 8, θ = 0, n = 3 and k = 0, 1, 2.
Applying De Moivre gives us the roots: 2 (cos 0 + i sin 0), 2 (cos (2π/3) + i sin (2π/3)), 2 (cos (4π/3) + i sin (4π/3)) or in rectangular form 2, -1 + √3i, -1 - √3i.

Understanding the concept of nth roots of complex numbers enhances your grasp of algebra, geometry, and trigonometry.
It is fundamental to solving complex math problems across a range of disciplines including engineering, physics, and computer science, making it a crucial skill for further studies.
A solid understanding of how to perform complex number root extraction is vital for succeeding in further mathematics.