Polar Form of Complex Numbers: Learn It 5

Finding Powers and Roots of Complex Numbers in Polar Form

Finding powers of complex numbers is greatly simplified using De Moivre’s Theorem. It states that, for a positive integer [latex]n,{z}^{n}[/latex] is found by raising the modulus to the [latex]n\text{th}[/latex] power and multiplying the argument by [latex]n[/latex]. It is the standard method used in modern mathematics.

De Moivre’s Theorem

If [latex]z=r\left(\cos \theta +i\sin \theta \right)[/latex] is a complex number, then

[latex]\begin{align}&{z}^{n}={r}^{n}\left[\cos \left(n\theta \right)+i\sin \left(n\theta \right)\right]\\ &{z}^{n}={r}^{n}\text{cis}\left(n\theta \right)\end{align}[/latex]

where [latex]n[/latex] is a positive integer.

Evaluate the expression [latex]{\left(1+i\right)}^{5}[/latex] using De Moivre’s Theorem.

Finding Roots of Complex Numbers in Polar Form

To find the nth root of a complex number in polar form, we use the [latex]n\text{th}[/latex] Root Theorem or De Moivre’s Theorem and raise the complex number to a power with a rational exponent. There are several ways to represent a formula for finding [latex]n\text{th}[/latex] roots of complex numbers in polar form.

the nth root theorem

To find the [latex]n\text{th}[/latex] root of a complex number in polar form, use the formula given as

[latex]\begin{align}{z}^{\frac{1}{n}}={r}^{\frac{1}{n}}\left[\cos \left(\frac{\theta }{n}+\frac{2k\pi }{n}\right)+i\sin \left(\frac{\theta }{n}+\frac{2k\pi }{n}\right)\right]\end{align}[/latex]

where [latex]k=0,1,2,3,...,n - 1[/latex]. We add [latex]\frac{2k\pi }{n}[/latex] to [latex]\frac{\theta }{n}[/latex] in order to obtain the periodic roots.

Evaluate the cube roots of [latex]z=8\left(\cos \left(\frac{2\pi }{3}\right)+i\sin \left(\frac{2\pi }{3}\right)\right)[/latex].

Find the four fourth roots of [latex]16\left(\cos \left(120^\circ \right)+i\sin \left(120^\circ \right)\right)[/latex].