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The question in the image asks to find the value of \( i^{40} \), where \( i \) is the imaginary unit defined as \( i = \sqrt{-1} \).

The powers of \( i \) follow a cyclic pattern with a period of 4:

- \( i^1 = i \)
- \( i^2 = -1 \)
- \( i^3 = -i \)
- \( i^4 = 1 \)

This cycle repeats every four powers, so we can calculate \( i^{40} \) by reducing the exponent modulo 4:

\[
40 \mod 4 = 0
\]

Thus, \( i^{40} \) corresponds to \( i^0 = 1 \).

Therefore, the value of \( i^{40} \) is \( 1 \), not \( -1 \) as listed in the options.

Would you like more details or have any questions?

Here are some related questions:
1. What is the value of \( i^{45} \)?
2. What are the first four powers of \( i \)?
3. How can we calculate the value of \( i^{n} \) for any integer \( n \)?
4. Why does \( i^4 = 1 \) and how does the cyclic nature work?
5. What is the application of imaginary numbers in electrical engineering?

**Tip:** Always reduce powers of \( i \) modulo 4 to simplify calculations!

Discover how to solve for the value of i^40 by understanding the cyclic nature of powers of the imaginary unit i. This problem focuses on complex numbers and algebraic concepts, suitable for high school students.

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