The image shows a request for the fourth roots of $16i$, specifically asking for the value of $(16i)^{1/4}$.

Let's break this down:

Step 1: Express $16i$ in polar form

We need to express the complex number $16i$ in polar form:

• $16i$ is purely imaginary, with real part $0$ and imaginary part $16$.
• The magnitude $|z|$ of $16i$ is: $|z| = \sqrt{0^2 + 16^2} = 16$
• The argument (angle) $\theta$ of $16i$ is the angle it makes with the positive real axis. Since $16i$ lies on the positive imaginary axis, the argument is: $\theta = \frac{\pi}{2}$

Thus, the polar form of $16i$ is: $16i = 16 \text{cis} \frac{\pi}{2}$ where $\text{cis} \theta$ stands for $\cos \theta + i \sin \theta$.

Step 2: Find the fourth roots

To find the fourth roots of a complex number, we use De Moivre's Theorem. The general formula for the $n$-th roots of a complex number is: $z^{1/n} = r^{1/n} \left( \text{cis} \left( \frac{\theta + 2k\pi}{n} \right) \right)$ where $r$ is the magnitude, $\theta$ is the argument, $n$ is the root to be taken (in this case $4$), and $k$ is an integer representing different roots.

For $n = 4$, we get four distinct roots corresponding to $k = 0, 1, 2, 3$. The magnitude of each root is: $r^{1/4} = 16^{1/4} = 2$ Now, we compute the arguments for each root: $\theta_k = \frac{\pi}{2} + 2k\pi$ Then divide by $4$: $\theta_k = \frac{\pi}{8} + \frac{k\pi}{2}, \quad k = 0, 1, 2, 3$

Step 3: Calculate the four roots

The four roots are:

1. $2 \text{cis} \frac{\pi}{8}$
2. $2 \text{cis} \frac{5\pi}{8}$
3. $2 \text{cis} \frac{9\pi}{8}$
4. $2 \text{cis} \frac{13\pi}{8}$

These are the four distinct fourth roots of $16i$.

Would you like further details or clarification on any of the steps?

Here are five related questions for practice:

1. How would you find the cube roots of $27i$?
2. What is the polar form of the complex number $4 - 4i$?
3. How do you find the roots of a complex number with negative magnitude?
4. What is De Moivre's Theorem and how is it applied in finding complex roots?
5. How can you convert a complex number from polar to rectangular form?

Tip: When solving for roots of complex numbers, always remember to account for all distinct angles (using $2k\pi$) to capture all possible roots!