6.5: De Moivre's and the nth Root Theorem

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

DE MOIVRE’S THEOREM

If \(z=r(\cos \theta+i \sin \theta)\) is a complex number, then

\[\begin z^n &= r^n[\cos(n\theta)+i \sin(n\theta) ] \\ z^n &= r^n\space cis(n\theta) \end\]

where \(n\) is a positive integer.

Example \(\PageIndex\): Evaluating an Expression Using De Moivre’s Theorem

Evaluate the expression \(^5\) using De Moivre’s Theorem.

Solution

Since De Moivre’s Theorem applies to complex numbers written in polar form, we must first write \((1+i)\) in polar form. Let us find \(r\).

Then we find \(\theta\). Using the formula \(\tan \theta=\dfrac\) gives

\[\begin \tan \theta &= \dfrac \\ \tan \theta &= 1 \\ \theta &= \dfrac<\pi> \end\]

Use De Moivre’s Theorem to evaluate the expression.

Finding Roots of Complex Numbers in Polar Form

To find the \(n^\) root of a complex number in polar form, we use the \(n^\) 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 \(n^\) roots of complex numbers in polar form.

THE \(N^\) ROOT THEOREM

To find the \(n^\) root of a complex number in polar form, use the formula given as

where \(k=0, 1, 2, 3, . . . , n−1\). We add \(\dfrac<2k\pi>\) to \(\dfrac\) in order to obtain the periodic roots.

Example \(\PageIndex\): the Root of a Complex Number

Evaluate the cube roots of \(z=8\left(\cos\left(\frac<2\pi>\right)+i\sin\left(\frac<2\pi>\right)\right)\).

Solution

There will be three roots: \(k=0, 1, 2\). When \(k=0\), we have

When \(k=1\), we have

When \(k=2\), we have

Remember to find the common denominator to simplify fractions in situations like this one. For \(k=1\), the angle simplification is

Find the four fourth roots of \(16(\cos(120°)+i \sin(120°))\).

Extra Practice

For the following exercises, find the powers of each complex number in polar form.

1. Find \(z^\) when \(z=2\) cis \(\left(70^\right)\)
2. Find \(z^\) when \(z=5\) cis \(\left(\frac\right)\)

For the following exercises, evaluate each root.

1. Evaluate the cube root of \(z\) when \(z=64\) cis \(\left(210^\right)\).
2. Evaluate the square root of \(z\) when \(z=25\) cis \(\left(\frac\right)\).

Key Concepts

• To find the power of a complex number \(z^n\), raise \(r\) to the power \(n\), and multiply \(\theta\) by \(n\). See Example \(\PageIndex\).
• Finding the roots of a complex number is the same as raising a complex number to a power, but using a rational exponent. See Example \(\PageIndex\).

Contributors and Attributions

• Jay Abramson (Arizona State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at https://openstax.org/details/books/precalculus.

This page titled 6.5: De Moivre's and the nth Root Theorem is shared under a CK-12 license and was authored, remixed, and/or curated by CK-12 Foundation.

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