Question

Find all the complex roots. Write your answers in exponential form. The complex fourth roots of \(4-4 \sqrt{3} i\)

Short answer

The complex fourth roots are \(2e^{i(-\frac{\pi}{12})}\), \(2e^{i(\frac{5\pi}{12})}\), \(2e^{i(\frac{13\pi}{12})}\), and \(2e^{i(\frac{7\pi}{4})}\).

Step-by-step solution

Express the complex number in polar form

The complex number given is \(4 - 4\sqrt{3} i\). To express this in polar form, first find the magnitude \(r\) and argument \(\theta\). The magnitude is given by \(r = \sqrt{a^2 + b^2}\), where \(a = 4\) and \(b = -4\sqrt{3}\). Thus, \[ r = \sqrt{4^2 + (-4\sqrt{3})^2} = \sqrt{16 + 48} = \sqrt{64} = 8 \]. The argument is \(\theta = \arctan\left(\frac{b}{a}\right)\). Hence, \[ \theta = \arctan\left(\frac{-4\sqrt{3}}{4}\right) = \arctan(-\sqrt{3}) = -\frac{\pi}{3} \]. Therefore, the polar form of the complex number is \[ 8 \text{cis} \left(-\frac{\pi}{3}\right) \].

Use De Moivre's Theorem to find the fourth roots

The fourth roots of a complex number \(re^{i\theta}\) are given by \[ \sqrt[4]{r} \text{cis} \left(\frac{\theta + 2k\pi}{4}\right) \], for \(k = 0, 1, 2, 3\). First, compute \(\sqrt[4]{8} = \sqrt[4]{2^3 \cdot 2^3} = 2\).

Compute the roots for each value of k

Now determine the arguments for the four roots by substituting \(k = 0, 1, 2, 3\): \[ \text{For } k = 0: \theta_0 = \frac{-\frac{\pi}{3}}{4} = -\frac{\pi}{12} \] \[ \text{For } k = 1: \theta_1 = \frac{-\frac{\pi}{3} + 2\pi}{4} = \frac{5\pi}{12} \] \[ \text{For } k = 2: \theta_2 = \frac{-\frac{\pi}{3} + 4\pi}{4} = \frac{13\pi}{12} \] \[ \text{For } k = 3: \theta_3 = \frac{-\frac{\pi}{3} + 6\pi}{4} = \frac{21\pi}{12} = \frac{7\pi}{4} \]. The fourth roots in exponential form are then: \[ 2e^{i(-\frac{\pi}{12})}, \quad 2e^{i(\frac{5\pi}{12})}, \quad 2e^{i(\frac{13\pi}{12})}, \quad 2e^{i(\frac{7\pi}{4})} \]

Key concepts

Polar Form

To find the complex roots of a number, we often first convert it into **polar form**. Polar form represents a complex number using its magnitude and angle.
The given complex number is \(4 - 4 \sqrt{3} i\). We need to express this number in polar form.

First, we find the magnitude \(r\). This is done using the formula \( r = \sqrt{a^2 + b^2} \), where \(a\) is the real part and \(b\) is the imaginary part. For our number:

• \( a = 4 \)
• \( b = -4 \sqrt{3} \)

Therefore,
\[ r = \sqrt{4^2 + (-4 \sqrt{3})^2} = \sqrt{16 + 48} = \sqrt{64} = 8 \]
Next, we determine the argument \(\theta\). The argument is found using \( \theta = \text{arctan} \left( \frac{b}{a} \right) \):
\[ \theta = \text{arctan} \left( \frac{-4 \sqrt{3}}{4} \right) = \text{arctan}(-\sqrt{3}) = -\frac{\pi}{3} \]
Now, in polar form, the complex number is expressed as \( r \text{cis} \theta \):
\[ 8 \text{cis} \left(-\frac{\pi}{3}\right) \]

De Moivre's Theorem

De Moivre's Theorem is a powerful tool when working with complex numbers. It is especially useful for finding powers and roots of complex numbers.
According to De Moivre's Theorem, for a complex number in polar form \( r \text{cis} \theta \), the \( n \)-th roots are given by:
\[ \sqrt[n]{r} \text{cis} \left( \frac{\theta + 2k\pi}{n} \right) \]
Here, \( k \) represents the different roots, ranging from 0 to \( n-1 \).
For this example, we are looking for the fourth roots, so \( n = 4 \). De Moivre's Theorem helps us find the roots by simplifying the process of dealing with complex exponentiation. First, calculate the fourth root of the magnitude \( r = 8 \):
\[ \sqrt[4]{8} = \sqrt[4]{2^3 \cdot 2^3} = 2 \]
Then, for each \( k = 0, 1, 2, 3 \), calculate the argument:

• For \( k = 0 \): \( \frac{ -\frac{\pi}{3} + 2(0)\pi}{4} = -\frac{\pi}{12} \)
• For \( k = 1 \): \( \frac{ -\frac{\pi}{3} + 2(1)\pi}{4} = \frac{5\pi}{12} \)
• For \( k = 2 \): \( \frac{ -\frac{\pi}{3} + 2(2)\pi}{4} = \frac{13\pi}{12} \)
• For \( k = 3 \): \( \frac{ -\frac{\pi}{3} + 2(3)\pi}{4} = \frac{7\pi}{4} \)

Magnitude and Argument

The magnitude and argument are foundational concepts in working with complex numbers in polar form. The **magnitude** represents the distance of the complex number from the origin in the complex plane. In mathematical terms, it is defined as:
\[ r = \sqrt{a^2 + b^2} \]
For the number \(4 - 4 \sqrt{3} i\):
\[ r = \sqrt{4^2 + (-4 \sqrt{3})^2} = \sqrt{16 + 48} = \sqrt{64} = 8 \]
The **argument** of a complex number is the angle it makes with the positive real axis. It can be found using arctangent:
\[ \theta = \text{arctan} \left( \frac{b}{a} \right) \]
For our number:
\[ \theta = \text{arctan} \left( \frac{-4 \sqrt{3}}{4} \right) = \text{arctan}(-\sqrt{3}) = -\frac{\pi}{3} \]

Knowing both the magnitude and the argument allows us to express any complex number in polar form, which is incredibly useful for operations like multiplication, division, and finding roots.

Fourth Roots

Finding the fourth roots of a complex number can be streamlined using De Moivre's Theorem. For a number in polar form \( r \text{cis} \theta \), the fourth roots are:
\[ \sqrt[4]{r} \text{cis} \left( \frac{\theta + 2k\pi}{4} \right) \]
Converting our example number, \( 4 - 4 \sqrt{3} i\), into polar form gives us:
\( 8 \text{cis} \left(-\frac{\pi}{3} \right) \). We computed:

• \( \sqrt[4]{8} = 2 \)
• And the arguments for \( k = 0, 1, 2, 3 \)

This results in the following roots:

• \( 2e^{i(-\frac{\pi}{12})} \)
• \( 2e^{i(\frac{5\pi}{12})} \)
• \( 2e^{i(\frac{13\pi}{12})} \)
• \( 2e^{i(\frac{7\pi}{4})} \)

Each expression represents one of the four unique fourth roots of the original complex number.