Here is the diagram of the complex plane showing the five solutions of the equation \( z^5 = 32i \). The solutions are plotted on a circle of radius 2, with the points spaced evenly at angles of \( \frac{2\pi}{5} \) apart. Each point corresponds to one of the solutions \( z_0, z_1, z_2, z_3, z_4 \).

Would you like more details about the solutions or further explanations? Here are some related questions you might find interesting:

1. How can we derive the general form for the roots of any complex number equation like \( z^n = C \)?
2. What is the significance of the angle and radius in polar form for complex numbers?
3. How do we convert between rectangular and polar forms of complex numbers?
4. What happens if the equation has negative or fractional powers, and how do we solve those?
5. How does the symmetry of the unit circle relate to the solutions of complex equations?

**Tip**: When working with complex numbers, it's often helpful to think of them in terms of their magnitude (distance from the origin) and their angle (direction from the positive real axis). This makes operations like multiplication and division easier to visualize and calculate.

Learn how to solve the equation Z^5 = 32i by finding the five roots and plotting them on the complex plane. This problem covers concepts such as complex numbers, polar form, De Moivre's Theorem, and the symmetry of the unit circle. Suitable for students in Grades 11-12.

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