Raising a complex number to a natural power

In this publication, we will consider how a complex number can be raised to a power (including using the De Moivre formula). The theoretical material is accompanied by examples for better understanding.

Raising a complex number to a power

First, remember that a complex number has the general form: z = a + bi (algebraic form).

Now we can proceed directly to the solution of the problem.

Square number

We can represent the degree as a product of the same factors, and then find their product (while remembering that i² =-1).

z² = (a + bi)² = (a + bi)(a + bi)

Example 1:

z=3+5i

z² = (3 + 5i)² = (3 + 5i)(3 + 5i) = 9 + 15i + 15i + 25i² = -16 + 30i

You can also use, namely the square of the sum:

z² = (a + bi)² = a² + 2 ⋅ a ⋅ bi + (bi)² = a² + 2abi – b²

Note: In the same way, if necessary, formulas for the square of the difference, the cube of the sum / difference, etc. can be obtained.

Nth degree

Raise a complex number z in kind n much easier if it is represented in trigonometric form.

Recall that, in general, the notation of a number looks like this: z = |z| ⋅ (cos φ + i ⋅ sin φ).

For exponentiation, you can use De Moivre’s formula (so named after the English mathematician Abraham de Moivre):

zⁿ = | z |ⁿ ⋅ (cos(nφ) + i ⋅ sin(nφ))

The formula is obtained by writing in trigonometric form (the modules are multiplied, and the arguments are added).

Example 2

Raise a complex number z = 2 ⋅ (cos 35° + i ⋅ sin 35°) to the eighth degree.

Solution

z⁸ = 2⁸ ⋅ (cos(8 ⋅ 35°) + i ⋅ sin(8 ⋅ 35°)) = 256 ⋅ (cos 280° + i sin 280°).