Which of the following is equal to i^8?

Solution:

We need to simplify \( i^8 \). 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 \( i^8 \), we divide the exponent by 4 and look at the remainder:

\( 8 \div 4 = 2 \) with a remainder of \( 0 \).

Therefore, \( i^8 = i^4 \) (or any power of 4), which is equal to \( 1 \).

So, \( i^8 = 1 \).

Now let's compare this with the given options:

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

The value \( 1 \) matches option 4.

Answer: 4. 1