To solve the expression \( i^{-2i} \), let's break it down step by step.

### Step 1: Express \( i \) in exponential form
The complex number \( i \) can be represented in polar form as:
\[
i = e^{i\frac{\pi}{2}}
\]
### Step 2: Apply the exponentiation
Now we need to find \( i^{-2i} \). Substituting the exponential form:
\[
i^{-2i} = \left( e^{i\frac{\pi}{2}} \right)^{-2i}
\]
Using the property of exponents \( (a^b)^c = a^{bc} \):
\[
i^{-2i} = e^{-2i \cdot i\frac{\pi}{2}}
\]
### Step 3: Simplify the exponent
Simplifying the exponent \( -2i \cdot i\frac{\pi}{2} \):
\[
-2i \cdot i\frac{\pi}{2} = 2 \cdot \frac{\pi}{2} = \pi
\]
So the expression becomes:
\[
i^{-2i} = e^{\pi}
\]

Thus, the value of \( i^{-2i} \) is \( e^{\pi} \), which is approximately 23.1407.

Would you like more details or have any questions about this solution?

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Here are 5 related questions to expand your understanding:

1. What is the polar form of any other complex number \( a + bi \)?
2. How do you convert an expression involving \( e \) back to rectangular form?
3. What is the significance of Euler's formula in complex numbers?
4. How do you find the magnitude and angle of a complex number?
5. What is the value of \( i^i \)?

**Tip:** When dealing with complex exponents, converting the base to its exponential form often simplifies the calculations.

Explore how to solve the complex number exponentiation problem i^-2i step-by-step. Learn about complex numbers, exponential forms, and Euler's formula in this detailed solution. Suitable for advanced high school students and beyond.

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