Powers of Complex Numbers (DeMoivre's Theorem): Videos & Practice Problems

Complex numbers can be expressed in polar form, which is particularly useful for performing operations like multiplication and exponentiation. When multiplying complex numbers in polar form, the process involves taking the product of their magnitudes (r values) and adding their angles. However, a more efficient method for raising complex numbers to a power is through de Moivre's theorem.

According to de Moivre's theorem, to raise a complex number in polar form to a power \( n \), you follow these steps: take the magnitude \( r \) and raise it to the power \( n \), and then multiply the angle \( \theta \) by \( n \). This can be expressed mathematically as:

\[ (r \text{ cis } \theta)^n = r^n \text{ cis } (n \theta) \]

Here, "cis" is shorthand for \( \cos \theta + i \sin \theta \). For example, if you have a complex number represented as \( 3 \text{ cis } 15^\circ \) and you want to square it, you would calculate:

\[ 3^2 \text{ cis } (2 \times 15^\circ) = 9 \text{ cis } 30^\circ \]

This method not only simplifies calculations but also provides a quick way to find results without tedious multiplication. In another example, if you have a complex number \( 4 \text{ cis } \frac{\pi}{6} \) and you want to raise it to the third power, you would compute:

\[ 4^3 \text{ cis } (3 \times \frac{\pi}{6}) = 64 \text{ cis } \frac{\pi}{2} \]

In this case, \( 4^3 \) equals 64, and \( 3 \times \frac{\pi}{6} \) simplifies to \( \frac{\pi}{2} \). This demonstrates how de Moivre's theorem can significantly streamline the process of working with complex numbers, especially when dealing with higher powers.

Understanding and applying de Moivre's theorem is essential for efficiently solving problems involving complex numbers in polar form, making it a valuable tool in both theoretical and practical applications.

To solve the problem of dividing two complex numbers in polar form and raising the result to the third power, we can break the process into two main steps: division and exponentiation. Let's denote the two complex numbers as \( z_1 \) and \( z_2 \), represented in polar form as \( r_1 \text{cis} \theta_1 \) and \( r_2 \text{cis} \theta_2 \), respectively.

In the first step, we need to find the quotient \( \frac{z_1}{z_2} \). The rules for dividing complex numbers in polar form state that we divide the magnitudes (r values) and subtract the angles. For our example, if \( r_1 = 4 \) and \( r_2 = \frac{1}{2} \), we calculate:

\[\frac{r_1}{r_2} = \frac{4}{\frac{1}{2}} = 4 \times 2 = 8\]

Next, we subtract the angles: if \( \theta_1 = 25^\circ \) and \( \theta_2 = 10^\circ \), we have:

\[\theta_1 - \theta_2 = 25^\circ - 10^\circ = 15^\circ\]

Thus, the result of the division is:

\[\frac{z_1}{z_2} = 8 \text{cis} 15^\circ\]

In the second step, we raise this result to the third power using De Moivre's theorem, which states that \( (r \text{cis} \theta)^n = r^n \text{cis} (n\theta) \). Therefore, we compute:

\[\left(8 \text{cis} 15^\circ\right)^3 = 8^3 \text{cis} (3 \times 15^\circ)\]

Calculating \( 8^3 \) gives:

\[8^3 = 512\]

And for the angle:

\[3 \times 15^\circ = 45^\circ\]

Combining these results, we find:

\[\left( \frac{z_1}{z_2} \right)^3 = 512 \text{cis} 45^\circ\]

This final expression represents the solution to the problem, demonstrating how to handle multiple operations involving complex numbers in polar form by systematically addressing division first and then exponentiation.

Understanding how to find the roots of complex numbers in polar form is essential for mastering complex analysis. When dealing with roots, such as square roots or cube roots, the process can initially seem tedious, but it becomes manageable with practice. The fundamental concept is based on de Moivre's theorem, which allows us to express complex numbers in polar form and manipulate them effectively.

To find the nth root of a complex number expressed as \( r \text{cis} \theta \), where \( r \) is the modulus and \( \theta \) is the argument, we use the formula:

$$ z_k = r^{\frac{1}{n}} \text{cis} \left( \frac{\theta + 360^\circ k}{n} \right) $$

Here, \( k \) takes on integer values from \( 0 \) to \( n-1 \), which gives us all possible roots. The term \( 360^\circ k \) accounts for the periodic nature of the complex plane, allowing us to find multiple solutions by rotating around the circle.

For example, to find the cube root of the complex number \( 8 \text{cis} 45^\circ \), we first calculate the modulus raised to the power of \( \frac{1}{3} \): \( 8^{\frac{1}{3}} = 2 \). Next, we determine the angles for each root:

1. For \( k = 0 \):

$$ \theta_0 = \frac{45^\circ + 360^\circ \cdot 0}{3} = \frac{45^\circ}{3} = 15^\circ $$

2. For \( k = 1 \):

$$ \theta_1 = \frac{45^\circ + 360^\circ \cdot 1}{3} = \frac{405^\circ}{3} = 135^\circ $$

3. For \( k = 2 \):

$$ \theta_2 = \frac{45^\circ + 360^\circ \cdot 2}{3} = \frac{765^\circ}{3} = 255^\circ $$

Thus, the three cube roots of \( 8 \text{cis} 45^\circ \) are:

$$ z_0 = 2 \text{cis} 15^\circ, \quad z_1 = 2 \text{cis} 135^\circ, \quad z_2 = 2 \text{cis} 255^\circ $$

In summary, finding the roots of complex numbers involves calculating the modulus raised to the appropriate power and determining the angles by considering the periodic nature of the complex plane. With practice, this process becomes straightforward, allowing for a deeper understanding of complex number operations.

To find the 5th root of the complex number \( z = 1024 \cdot \text{cis}\left(\frac{\pi}{2}\right) \), we start by recognizing that the 5th root can be expressed as \( z^{\frac{1}{5}} \). In polar form, the 5th root of a complex number is calculated using the formula:

\[z^{\frac{1}{n}} = r^{\frac{1}{n}} \cdot \text{cis}\left(\frac{\theta + 2\pi k}{n}\right)\]

Here, \( r \) is the modulus (magnitude) of the complex number, \( \theta \) is the argument (angle), and \( k \) takes on integer values from \( 0 \) to \( n-1 \). In this case, \( r = 1024 \) and \( \theta = \frac{\pi}{2} \), with \( n = 5 \).

First, we calculate the modulus for the 5th root:

\[r^{\frac{1}{5}} = 1024^{\frac{1}{5}} = 4\]

Next, we determine the angles for each of the 5 roots using the formula for \( \theta_k \):

\[\theta_k = \frac{\theta + 2\pi k}{n}\]

Substituting the known values, we find:

\[\theta_k = \frac{\frac{\pi}{2} + 2\pi k}{5}\]

Now, we calculate \( \theta_k \) for \( k = 0, 1, 2, 3, 4 \):

For \( k = 0 \):

\[\theta_0 = \frac{\frac{\pi}{2} + 2\pi \cdot 0}{5} = \frac{\pi}{10}\]

For \( k = 1 \):

\[\theta_1 = \frac{\frac{\pi}{2} + 2\pi \cdot 1}{5} = \frac{\frac{\pi}{2} + 2\pi}{5} = \frac{\frac{\pi}{2} + \frac{4\pi}{2}}{5} = \frac{5\pi}{10} = \frac{\pi}{2}\]

For \( k = 2 \):

\[\theta_2 = \frac{\frac{\pi}{2} + 2\pi \cdot 2}{5} = \frac{\frac{\pi}{2} + 4\pi}{5} = \frac{\frac{\pi}{2} + \frac{8\pi}{2}}{5} = \frac{9\pi}{10}\]

For \( k = 3 \):

\[\theta_3 = \frac{\frac{\pi}{2} + 2\pi \cdot 3}{5} = \frac{\frac{\pi}{2} + 6\pi}{5} = \frac{\frac{\pi}{2} + \frac{12\pi}{2}}{5} = \frac{13\pi}{10}\]

For \( k = 4 \):

\[\theta_4 = \frac{\frac{\pi}{2} + 2\pi \cdot 4}{5} = \frac{\frac{\pi}{2} + 8\pi}{5} = \frac{\frac{\pi}{2} + \frac{16\pi}{2}}{5} = \frac{17\pi}{10}\]

Now we can express the 5th roots of \( z \) as:

\[z_k = 4 \cdot \text{cis}(\theta_k) \quad \text{for } k = 0, 1, 2, 3, 4\]

Thus, the five solutions are:

\[z_0 = 4 \cdot \text{cis}\left(\frac{\pi}{10}\right), \quad z_1 = 4 \cdot \text{cis}\left(\frac{\pi}{2}\right), \quad z_2 = 4 \cdot \text{cis}\left(\frac{9\pi}{10}\right), \quad z_3 = 4 \cdot \text{cis}\left(\frac{13\pi}{10}\right), \quad z_4 = 4 \cdot \text{cis}\left(\frac{17\pi}{10}\right)\]

These represent the five distinct 5th roots of the complex number \( z \).