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,