Which of the following is equal to i^11?

Solution:

We need to simplify \( i^{11} \). We know that the powers of \( i \) cycle with a period of 4:

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

To simplify \( i^{11} \), we divide the exponent by 4 and look at the remainder:

\( 11 \div 4 = 2 \) with a remainder of \( 3 \).

Therefore, \( i^{11} = i^3 \).

From the cycle of powers of \( i \), we know that \( i^3 = -i \).

So, \( i^{11} = -i \).

Now let's compare this with the given options:

• 1. \( -1 \)
• 2. \( i \)
• 3. \( -i \)
• 4. \( 1 \)

The value \( -i \) matches option 3.

Answer: 3. -i