Powers and roots of complex numbers are easiest to understand in polar form. If:
\[
z=a+bi
\]
then \(z\) can be written as:
\[
z=re^{i\theta}
\]
where:
\[
r=|z|=\sqrt{a^2+b^2},
\qquad
\theta=\arg(z)
\]
The modulus \(r\) measures the distance from the origin, and the argument \(\theta\) measures direction.
1. De Moivre’s theorem
De Moivre’s theorem says:
\[
(re^{i\theta})^m=r^m e^{im\theta}
\]
This means that powers of complex numbers scale the length and multiply the angle.
In trigonometric form:
\[
\left(r(\cos\theta+i\sin\theta)\right)^m
=
r^m(\cos(m\theta)+i\sin(m\theta))
\]
2. Integer powers
To compute \(z^m\), first write \(z=re^{i\theta}\), then apply:
\[
z^m=r^m e^{im\theta}
\]
For example, if \(z=1+i\), then:
\[
|z|=\sqrt2,\qquad \arg(z)=\frac{\pi}{4}
\]
Therefore:
\[
(1+i)^5=(\sqrt2)^5e^{i5\pi/4}
\]
The graph shows the final value as a vector on the complex plane.
3. \(n\)-th roots of a complex number
The \(n\)-th roots of \(z\) are the solutions of:
\[
w^n=z
\]
If \(z=re^{i\theta}\), then all \(n\) roots are:
\[
w_k=r^{1/n}e^{i(\theta+2\pi k)/n},
\qquad
k=0,1,2,\ldots,n-1
\]
These roots all have the same modulus \(r^{1/n}\), so they lie on a circle centered at the origin.
They are equally spaced by:
\[
\frac{2\pi}{n}
\]
4. Example: cube roots of \(-8\)
Write \(-8\) in polar form:
\[
-8=8e^{i\pi}
\]
The cube roots satisfy:
\[
w^3=-8
\]
The formula gives:
\[
w_k=8^{1/3}e^{i(\pi+2\pi k)/3}
=
2e^{i(\pi+2\pi k)/3}
\]
for \(k=0,1,2\). The three roots are:
\[
1+\sqrt3\,i,\qquad -2,\qquad 1-\sqrt3\,i
\]
All three are shown on a circle of radius \(2\).
5. Rational powers
A rational power such as \(z^{p/q}\) is interpreted using roots:
\[
w=z^{p/q}
\quad\Longleftrightarrow\quad
w^q=z^p
\]
This usually gives \(q\) possible complex values, not just one. That is why the calculator lists all branches.
6. Why roots are equally spaced
Complex angles repeat every \(2\pi\). So the same number \(z\) can be written using angles:
\[
\theta,\quad \theta+2\pi,\quad \theta+4\pi,\ldots
\]
When taking \(n\)-th roots, these angles are divided by \(n\), producing \(n\) different directions.
That is why the roots form a regular pattern on a circle.
7. Formula summary
| Concept |
Formula |
Meaning |
| Polar form |
\(z=re^{i\theta}\) |
Length and direction |
| Integer power |
\(z^m=r^m e^{im\theta}\) |
Raise modulus, multiply angle |
| \(n\)-th root |
\(w_k=r^{1/n}e^{i(\theta+2\pi k)/n}\) |
All roots of \(w^n=z\) |
| Root spacing |
\(\Delta\theta=2\pi/n\) |
Equal angular separation |
| Rational power |
\(w^q=z^p\) |
All values of \(z^{p/q}\) |
8. Common mistakes
- Do not give only one root when an \(n\)-th root usually has \(n\) complex values.
- Do not forget the \(2\pi k\) term in the root formula.
- Negative powers of \(0\) are undefined.
- Use polar form for roots; rectangular form alone can hide the geometry.
- All \(n\)-th roots of the same number lie on the same circle.
- Rational powers like \(z^{1/3}\) are multi-valued in the complex plane.
