To solve the equation \( z^4 = 16i \), we need to express it in polar form and find the solutions for \( z \).

### Step 1: Express \( 16i \) in polar form
We can write \( 16i \) in polar form. Recall that the complex number \( 16i \) is located on the imaginary axis, 16 units above the origin. This corresponds to an argument \( \theta \) of \( \frac{\pi}{2} \).

So, we can write:
\[
16i = 16 \left( \cos \frac{\pi}{2} + i \sin \frac{\pi}{2} \right)
\]

### Step 2: Use De Moivre's Theorem
The equation \( z^4 = 16i \) can now be rewritten as:
\[
z^4 = 16 \left( \cos \frac{\pi}{2} + i \sin \frac{\pi}{2} \right)
\]

Taking the fourth root on both sides, we use De Moivre’s Theorem:
\[
z = \sqrt[4]{16} \left( \cos \frac{\frac{\pi}{2} + 2k\pi}{4} + i \sin \frac{\frac{\pi}{2} + 2k\pi}{4} \right)
\]
where \( k = 0, 1, 

Explore how to solve the complex equation Z^4 = 16i by converting to polar form and applying De Moivre's Theorem. This solution provides a detailed step-by-step approach, suitable for advanced high school students in Grades 11-12.

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