The nth root of a complex number is one of the n distinct complex numbers which, when raised to the nth power, gives the original complex number. It can be calculated using polar form and De Moivre's Theorem.
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The nth roots of a complex number are evenly spaced around a circle in the complex plane.
To find the nth root, convert the complex number to its polar form.
Use De Moivre's Theorem: $z^{1/n} = r^{1/n} \left(\cos\left(\frac{\theta + 2k\pi}{n}\right) + i \sin\left(\frac{\theta + 2k\pi}{n}\right)\right)$ for $k = 0, 1, ..., n-1$.
There are exactly n distinct nth roots for any non-zero complex number.
For $k=0$, the principal nth root is often used in applications.
Review Questions
How do you use polar form to find the nth roots of a complex number?
What is De Moivre's Theorem and how does it apply to finding nth roots?
Why are there exactly n distinct nth roots for any non-zero complex number?
Related terms
Polar Form: A way of representing a complex number in terms of its magnitude and angle: $z = r (\cos \theta + i \sin \theta)$.
De Moivre's Theorem: A formula that connects complex numbers and trigonometry: $(r (\cos \theta + i \sin \theta))^n = r^n (\cos(n \theta) + i \sin(n \theta))$.
Complex Plane: A two-dimensional plane where each point represents a complex number with real and imaginary parts as coordinates.