How to use De Moivre’s theorem for powers of complex numbers

Multiplying complex numbers in polar form multiplies their moduli and adds their angles. Raising one number to a power repeats that multiplication. De Moivre’s theorem collects the result into a formula we can use directly.

Apply De Moivre’s theorem

If $z = r (\cos θ + i \sin θ)$ , then for a positive integer $n$ ,

z^{n} = r^{n} (\cos (n θ) + i \sin (n θ))

The modulus becomes $r^{n}$ and the angle becomes $n θ$ . The formula also works for negative integer powers when $z \neq 0$ .

Cube a number already in polar form

Let $z = 7 (\cos (π / 6) + i \sin (π / 6))$ .

\begin{array}{l} z^{3} = 7^{3} (\cos (3 π / 6) + i \sin (3 π / 6)) \\ = 343 (\cos (π / 2) + i \sin (π / 2)) \\ = 343 i \end{array}

Raise a fractional modulus to a power

For $z = \frac{1}{2} (\cos (π / 7) + i \sin (π / 7))$ ,

z^{8} = \frac{1}{256} (\cos (8 π / 7) + i \sin (8 π / 7))

Convert rectangular form before taking a power

For $z = 1 - \sqrt{3} i$ , the modulus is $\sqrt{1 + 3} = 2$ and an argument is $- π / 3$ . To find its fourth power,

\begin{array}{l} z^{4} = 16 (\cos (- 4 π / 3) + i \sin (- 4 π / 3)) \\ = 16 (- \frac{1}{2} + \frac{\sqrt{3}}{2} i) \\ = - 8 + 8 \sqrt{3} i \end{array}

Use the equivalent exponential form

For $z = r e^{i θ}$ , the same rule is $z^{n} = r^{n} e^{i n θ}$ . If $r = 3$ , $θ = 3 π / 8$ , and $n = 12$ , we get $z^{12} = 531441 e^{9 π i / 2} = 531441 i$ .

Learn mathKrista KingAugust 18, 2021math, learn online, online course, online math, equilibrium solutions