De Moivre's Theorem: Powers and Roots
Raising a complex number to the 8th power by multiplying it out eight times is slow and error prone. Once the number is in trig form, a single rule replaces all that work: raise the modulus to the power and multiply the angle by the power. The same rule run in reverse finds every root, and a degree-n root has exactly n answers spaced evenly around a circle.
By the end of this page you can:
- Raise a complex number in trig form to an integer power using De Moivre’s theorem.
- Find all n distinct nth roots of a complex number.
- Place those roots correctly as evenly spaced points on a circle.
The floor. To take the nth power, raise the modulus to the nth power and multiply the angle by n. To take the nth roots, take the real nth root of the modulus and divide the angle by n, then step the angle by 360 degrees over n to land each of the n roots.
Quick Reference
| Field | Value |
|---|---|
| Textbook | Stewart, Redlin, Watson, Precalculus 7e |
| Chapter | 8 |
| Section | 8.3 Polar Form of Complex Numbers; De Moivre’s Theorem |
| Open alternate | OpenStax Precalculus 2e, 10.5 Polar Form of Complex Numbers |
| Difficulty | core |
| Time | 30 minutes |
Before You Start
The Idea
A complex number in trig form is z = r(cos theta + i sin theta), where r is the modulus (distance from the origin) and theta is the argument (angle from the positive real axis). De Moivre’s theorem says that taking a power scales the modulus by that power and rotates the angle by that power. Geometrically, raising to the nth power stretches the radius to r^n and multiplies the turn by n.
Roots reverse the process. Because every full turn of 360 degrees returns to the same point, a single complex number has n different nth roots. They all share the modulus r^(1/n) and their angles start at theta/n and then step around by 360/n degrees, so they sit as n equally spaced points on a circle of radius r^(1/n).
Key Formulas
De Moivre’s theorem (powers), for any positive integer n: \[ \bigl[r(\cos\theta + i\sin\theta)\bigr]^{n} = r^{n}\bigl(\cos n\theta + i\sin n\theta\bigr) \]
The n distinct nth roots of z = r(cos theta + i sin theta), for k = 0, 1, ..., n-1: \[ w_k = r^{1/n}\left(\cos\frac{\theta + 360^{\circ}k}{n} + i\sin\frac{\theta + 360^{\circ}k}{n}\right) \]
Source: Stewart 7e, Section 8.3 Polar Form of Complex Numbers; De Moivre’s Theorem.
Worked Example
Find all three cube roots of z = -8, and write each in standard a + bi form.
Step 1. Put z in trig form. The point -8 sits on the negative real axis, so r = 8 and theta = 180 degrees: z = 8(cos 180 + i sin 180).
Step 2. Set up the root formula with n = 3. The modulus of each root is 8^(1/3) = 2. The angles are (180 + 360k)/3 for k = 0, 1, 2.
Step 3. Step through k.
- k = 0: angle = 180/3 = 60 degrees.
- k = 1: angle = (180 + 360)/3 = 180 degrees.
- k = 2: angle = (180 + 720)/3 = 300 degrees.
Show the three roots
Each root has modulus 2, so evaluate 2(cos angle + i sin angle):
- k = 0: 2(cos 60 + i sin 60) = 2(1/2 + i(sqrt(3)/2)) = 1 + i*sqrt(3).
- k = 1: 2(cos 180 + i sin 180) = 2(-1 + 0i) = -2.
- k = 2: 2(cos 300 + i sin 300) = 2(1/2 - i(sqrt(3)/2)) = 1 - i*sqrt(3).
The three cube roots are 1 + isqrt(3), -2, and 1 - isqrt(3). They lie on a circle of radius 2, spaced 120 degrees apart. Check: cubing -2 gives -8, as required.
Connections
This builds directly on writing and multiplying complex numbers in trig form, since De Moivre’s theorem reads its power and root rules straight off the modulus and argument. It extends the angle-sum idea behind multiplication: a power is repeated multiplication, so the angles add up to n times theta. The root version connects to evenly spaced points on a circle, the same geometry that appears in roots of unity and in plotting in polar coordinates. The two most common slips to watch are finding only one nth root instead of all n, and forgetting to divide the angle by n when taking roots.
Resources
- Stewart, Redlin, Watson, Precalculus 7e, Section 8.3 Polar Form of Complex Numbers; De Moivre’s Theorem (primary).
- OpenStax Precalculus 2e, Section 10.5 Polar Form of Complex Numbers (alternate).
Last updated: 2026-06-24