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The expression given is \( e^{\frac{3\pi}{2} i} \). To solve this, we use **Euler's formula**:

\[
e^{ix} = \cos(x) + i \sin(x)
\]

Substituting \( x = \frac{3\pi}{2} \), we get:

\[
e^{\frac{3\pi}{2} i} = \cos\left(\frac{3\pi}{2}\right) + i \sin\left(\frac{3\pi}{2}\right)
\]

Now calculate the values of cosine and sine at \( \frac{3\pi}{2} \):

\[
\cos\left(\frac{3\pi}{2}\right) = 0
\]
\[
\sin\left(\frac{3\pi}{2}\right) = -1
\]

Thus:

\[
e^{\frac{3\pi}{2} i} = 0 + i(-1) = -i
\]

The correct answer is \( \boxed{-i} \), which corresponds to option **b**.

Would you like further details on this, or do you have any questions?

Here are 5 related questions for deeper understanding:
1. How is Euler's formula applied in other complex exponentials?
2. What is the geometric interpretation of \( e^{ix} \) on the complex plane?
3. How does the unit circle relate to sine and cosine values for different angles?
4. Can you derive the values of \( e^{i\pi} \) and \( e^{2\pi i} \) using Euler's formula?
5. What are the properties of the imaginary unit \( i \)?

**Tip**: Familiarize yourself with common angles like \( \pi/2, \pi, 3\pi/2 \), and \( 2\pi \) on the unit circle to quickly solve problems involving Euler's formula.

Learn how to solve the complex exponential expression e^(3π/2 * i) using Euler's formula. This detailed step-by-step solution covers the key concepts of complex numbers, Euler's formula, and trigonometric functions. Suitable for high school students in Grades 10-12.

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