Complex Number Roots: Understanding the Problem

The problem asks us to identify which of the given options cannot be a value of \(w\), given the equation \(w^4 = 16j\). This involves finding the fourth roots of the complex number \(16j\).

To find the roots of a complex number, it's generally easiest to first convert the complex number into its polar (or exponential) form. The general form of a complex number in exponential form is \(re^{j\theta}\), where \(r\) is the magnitude and \(\theta\) is the argument (angle).

Converting \(16j\) to Polar Form

First, let's express the given complex number \(16j\) in polar form.

• Magnitude (\(r\)): For \(16j\), which can be written as \(0 + 16j\), the magnitude \(r\) is given by \(\sqrt{0^2 + 16^2} = \sqrt{256} = 16\).
• Argument (\(\theta\)): The complex number \(16j\) lies on the positive imaginary axis in the complex plane. Therefore, its argument \(\theta\) is \(\frac{\pi}{2}\) radians.

So, \(16j\) in polar form can be written as \(16e^{j(\frac{\pi}{2} + 2k\pi)}\), where \(k\) is an integer. The \(2k\pi\) term accounts for all possible coterminal angles.

Finding the Fourth Roots of \(16j\)

We need to find \(w\) such that \(w^4 = 16j\). Using the polar form, we have:

\[ w^4 = 16e^{j(\frac{\pi}{2} + 2k\pi)} \]

To find \(w\), we take the fourth root of both sides:

\[ w = \left(16e^{j(\frac{\pi}{2} + 2k\pi)}\right)^{\frac{1}{4}} \]

Using De Moivre's Theorem for roots, we apply the power to both the magnitude and the argument:

\[ w = 16^{\frac{1}{4}} \cdot e^{j\left(\frac{\frac{\pi}{2} + 2k\pi}{4}\right)} \]

Simplify the terms:

\[ w = 2 \cdot e^{j\left(\frac{\pi}{8} + \frac{2k\pi}{4}\right)} \]

\[ w = 2 \cdot e^{j\left(\frac{\pi}{8} + \frac{k\pi}{2}\right)} \]

Calculating Distinct Values of \(w\)

To find the distinct fourth roots, we substitute integer values for \(k\), typically starting from \(k = 0, 1, 2, 3\).

• For \(k = 0\):

  \[ w_0 = 2e^{j\left(\frac{\pi}{8} + \frac{0\pi}{2}\right)} = 2e^{j\frac{\pi}{8}} \]

• For \(k = 1\):

  \[ w_1 = 2e^{j\left(\frac{\pi}{8} + \frac{1\pi}{2}\right)} = 2e^{j\left(\frac{\pi}{8} + \frac{4\pi}{8}\right)} = 2e^{j\frac{5\pi}{8}} \]

• For \(k = 2\):

  \[ w_2 = 2e^{j\left(\frac{\pi}{8} + \frac{2\pi}{2}\right)} = 2e^{j\left(\frac{\pi}{8} + \pi\right)} = 2e^{j\left(\frac{\pi}{8} + \frac{8\pi}{8}\right)} = 2e^{j\frac{9\pi}{8}} \]

• For \(k = 3\):

  \[ w_3 = 2e^{j\left(\frac{\pi}{8} + \frac{3\pi}{2}\right)} = 2e^{j\left(\frac{\pi}{8} + \frac{12\pi}{8}\right)} = 2e^{j\frac{13\pi}{8}} \]

Comparing with the Given Options

Now, let's compare our calculated distinct roots with the provided options:

The four possible values for \(w\) are \(2e^{j\frac{\pi}{8}}\), \(2e^{j\frac{5\pi}{8}}\), \(2e^{j\frac{9\pi}{8}}\), and \(2e^{j\frac{13\pi}{8}}\). Option 1, \(2e^{j\frac{2\pi}{8}}\) (which simplifies to \(2e^{j\frac{\pi}{4}}\)), is not among these calculated roots. Therefore, \(2e^{j\frac{2\pi}{8}}\) cannot be a value of \(w\).