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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!

Discover how to calculate the fourth roots of 16i using polar form and De Moivre's Theorem. This step-by-step solution covers key concepts in complex numbers and roots. Suitable for students in Grades 10-12.

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