Question

Find one or more values of each of the following complex expressions and compare with a computer solution. $$\sqrt[5]{-4-4 i}$$

Short answer

The approximate values for the expression are the fifth roots of (-4 - 4i) calculated using roots values.

Step-by-step solution

Represent the complex number in polar form

First, convert the complex number (-4 - 4i) into its polar form. The polar form of a complex number is given by \( re^{i\theta} \), where \( r \) is the magnitude and \( \theta \) is the argument (angle).

Calculate the magnitude

The magnitude \( r \) is found using the formula \( r = \sqrt{a^2 + b^2} \), where \( a \) and \( b \) are the real and imaginary parts of the complex number. For \( -4 - 4i \): \( r = \sqrt{(-4)^2 + (-4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \).

Determine the argument

The argument \( \theta \) is the angle the complex number makes with the positive real axis. It can be calculated using \( \tan^{-1}\left( \frac{b}{a} \right) \), but taking into account the quadrant. Since \( -4 - 4i \) is in the third quadrant, \( \theta = \tan^{-1}\left( \frac{-4}{-4} \right) + \pi = \frac{\pi}{4} + \pi = \frac{5\pi}{4} \).

Convert back to rectangular form

The complex number in polar form is \( 4\sqrt{2} e^{i\frac{5\pi}{4}} \). For the fifth root, apply De Moivre's Theorem, which states: \( \sqrt[n]{r} e^{i(\frac{\theta + 2k\pi}{n})} \) for all integers \( k \). So, we calculate \( \sqrt[5]{4\sqrt{2}} e^{i(\frac{5\pi + 2k\pi}{4}/5)} \) for k = 0, 1, 2, 3, 4.

Find all roots

Calculate the magnitude and argument of all five roots: \( r = (4\sqrt{2})^{1/5} \approx 1.3195 \) and\( \theta = \frac{(5\pi + 2k\pi)}{5} \text{ for } k = 0, 1, 2, 3, 4 \). This gives us five possible angles to calculate:

Calculate each root

For each k, compute the rectangular coordinates: \( z_k = 1.3195 e^{i\frac{5\pi + 2k\pi}{4}/5} \). Therefore, for k = 0, 1, 2, 3, 4, the values of z are: \( z_0 \approx 1.3195 e^{i \frac{5\pi}{4}} \).\( z_1 \approx 1.3195 e^{i \frac{5\pi + 2\pi}{4}} \cdots \).

Key concepts

complex number polar form

Complex numbers can be represented in a special way called the polar form. This form makes it easier to work with them, especially for multiplication and finding roots. The polar form of a complex number is given by \( re^{i\theta} \), where \( r \) is the magnitude of the complex number, and \( \theta \) is the argument (or angle).
To convert a complex number from rectangular form (i.e., \( a + bi \)) to polar form, follow these steps:

• Calculate the magnitude, \( r \), which is the distance from the origin to the point \( (a, b) \).
• Find the argument, \( \theta \), which is the angle the line from the origin to \( (a, b) \) makes with the positive real axis.

Using the given example, consider the complex number \( -4 - 4i \). Convert it to polar form by computing \( r \) and \( \theta \).

magnitude of complex numbers

The magnitude of a complex number is like its length or size. In mathematical terms, it is the distance from the origin to the point representing the complex number in the complex plane.
To find the magnitude of a complex number \( a + bi \), use the formula:
\[ r = \sqrt{a^2 + b^2} \]
For our example \( -4 - 4i \), the magnitude is:
\[ r = \sqrt{(-4)^2 + (-4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}\]
This means the distance from the origin to the point \( -4 - 4i \) is \( 4\sqrt{2} \).

argument of complex numbers

The argument of a complex number is the angle it makes with the positive real axis. This angle tells us the direction of the complex number.
To find the argument of a complex number \( a + bi \), you use the inverse tangent function, but you must also consider which quadrant the complex number lies in.
For the complex number \( -4 - 4i \), which is in the third quadrant, the angle is calculated as follows:
\[ \theta = \tan^{-1}\left(\frac{b}{a}\right) + \pi \]
Here, the argument \( \theta \) is:
\[ \theta = \frac{5\pi}{4}\]
Thus, the angle is \( \frac{5\pi}{4}\).

De Moivre's Theorem

De Moivre's Theorem is a powerful tool in working with complex numbers, especially for finding powers and roots. The theorem states:

\[ \left(re^{i\theta}\right)^n = r^n e^{in\theta}\]

This means to raise a complex number to a power, you raise the magnitude to that power and multiply the argument by that power.

For example, to find the fifth roots of \( -4 - 4i \) (or \( 4\sqrt{2}e^{i\frac{5\pi}{4}} \)), you can use the theorem to get:
\[ \sqrt[5]{4\sqrt{2}}e^{i\frac{5\pi}{4}\cdot\frac{1}{5} + 2k\pi/5}\]
This formula includes all the possible values (roots) for \( k = 0, 1, 2, 3, 4 \).

fifth roots of complex numbers

Finding the fifth roots of a complex number involves finding five separate values. Each root represents a different angle around the complex plane.
Given \( -4 - 4i \), converted to polar form as \( 4\sqrt{2} e^{i\frac{5\pi}{4}} \), we find the fifth roots by:

• Computing the fifth root of the magnitude: \( (4\sqrt{2})^{1/5} \approx 1.3195 \).
• Dividing the argument by 5 and adding \( 2k\pi/5 \) for each root.

For \( k = 0, 1, 2, 3, 4 \), the angles are:
\[ \frac{5\pi}{4} / 5 + \frac{2k\pi}{5}\]

Calculating each one, we get the five roots expressed in rectangular form. This process gives us complete values as the five distinct fifth roots of the original complex number.