We are given the equation:

\[
z^{\frac{3}{2}} = 5i
\]

To solve for \( z \), we need to isolate \( z \) by raising both sides of the equation to the power of \( \frac{2}{3} \). This will reverse the effect of raising \( z \) to the power of \( \frac{3}{2} \).

### Step 1: Raise both sides to the power of \( \frac{2}{3} \)
\[
\left( z^{\frac{3}{2}} \right)^{\frac{2}{3}} = (5i)^{\frac{2}{3}}
\]

On the left-hand side, \( \left( z^{\frac{3}{2}} \right)^{\frac{2}{3}} \) simplifies to \( z \), so we have:

\[
z = (5i)^{\frac{2}{3}}
\]

### Step 2: Simplify \( (5i)^{\frac{2}{3}} \)
To compute \( (5i)^{\frac{2}{3}} \), it's helpful to express \( 5i \) in polar form. The complex number \( 5i \) has a magnitude of 5 and an argument of \( \frac{\pi}{2} \) (since it's purely imaginary and points along the positive imaginary axis).

So, we can express \( 5i \) as:
\[
5i = 5 \cdot e^{i\frac{\pi}{2}}
\]

Now, we compute \( (5i)^{\frac{2}{3}} \) by raising the polar form to the power of \( \frac{2}{3} \):

\[
(5i)^{\frac{2}{3}} = \left( 5 \cdot e^{i\frac{\pi}{2}} \right)^{\frac{2}{3}} = 5^{\frac{2}{3}} \cdot e^{i\frac{2}{3} \cdot \frac{\pi}{2}} = 5^{\frac{2}{3}} \cdot e^{i\frac{\pi}{3}}
\]

### Step 3: Final Answer
Thus, the solution for \( z \) is:

\[
z = 5^{\frac{2}{3}} \cdot e^{i\frac{\pi}{3}}
\]

This represents a complex number in polar form. We can convert this back into rectangular form (a + bi) if needed, but this is the general solution.

Would you like me to express it in rectangular form, or do you have any other questions about this solution?

### Related Questions:
1. How do we convert complex numbers from polar to rectangular form?
2. What is the significance of the argument of a complex number in this type of problem?
3. How do we solve equations involving roots of complex numbers?
4. What is the geometric interpretation of raising a complex number to a fractional power?
5. How do we find all solutions to equations like this that involve complex numbers?

**Tip**: When dealing with complex numbers, the polar form is often the easiest way to work with powers and roots, especially when the argument is a simple fraction of \( \pi \).

Learn how to solve the equation z^{3/2} = 5i involving complex numbers and fractional powers. This solution uses polar form to simplify (5i)^{2/3} and covers key concepts in complex number theory. Ideal for students in Grades 11-12.

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