How Do You Find the Powers and Roots of a Complex Number?

Convert the complex number to polar form, then apply De Moivre’s theorem. For powers, raise the modulus to the power and multiply the argument by that power. For roots, take the corresponding root of the modulus and generate every valid argument.

This is IB Mathematics Analysis and Approaches HL content from AHL 1.14, suitable for exact non-calculator work on Paper 1.

The Method

Write a non-zero complex number in modulus-argument form:

$$z=r\operatorname{cis}\theta=r(\cos\theta+i\sin\theta).$$

Here, $r=|z|$ is the modulus, $\theta=\arg z$ is an argument, and $\operatorname{cis}\theta=\cos\theta+i\sin\theta$.

The formula works because multiplication in polar form multiplies moduli and adds arguments. Repeating the multiplication $n$ times therefore produces $r^n$ and $n\theta$. For roots, reversing this process requires dividing every angle equivalent to $\theta$, namely $\theta+2k\pi$, by . Restricting to lists each distinct root once. This explains both the formula and the equal spacing visible on an Argand diagram.

Worked Example

Let

$$z=2\operatorname{cis}\left(\frac{\pi}{3}\right).$$

For the fourth power, raise the modulus to four and multiply the argument by four:

$$z^4=2^4\operatorname{cis}\left(\frac{4\pi}{3}\right)=16\operatorname{cis}\left(\frac{4\pi}{3}\right).$$

For the three cube roots, substitute $n=3$ and use each integer value $k=0,1,2$:

$$w_k=\sqrt[3]{2}\operatorname{cis}\left(\frac{\frac{\pi}{3}+2k\pi}{3}\right).$$

The resulting arguments are $\frac{\pi}{9}$, $\frac{7\pi}{9}$, and . Each root has modulus .

Geometrically, the roots lie on a circle centred at the origin. Consecutive roots are equally spaced by $\frac{2\pi}{3}$, because dividing a full rotation of $2\pi$ among three roots gives that angular separation.

You can verify each answer by cubing it. The modulus becomes $2$, while each argument becomes equivalent to $\frac{\pi}{3}$ modulo $2\pi$.

A common misconception is giving only the principal root. If $z\neq0$, the equation $w^n=z$ has exactly $n$ distinct complex roots, so all values from to are required.

Exam technique: Show the polar form, state the correct formula, specify the values of $k$, and list every root exactly. A single root cannot earn full marks when the command term asks you to find all roots.