Questions: Simplify the expression. i^26 i^26=

Solution Steps

To simplify the expression \( i^{26} \), we need to use the properties of imaginary numbers. The imaginary unit \( i \) is defined as \( i = \sqrt{-1} \), and it has a cyclical pattern in its powers: \( i^1 = i \), \( i^2 = -1 \), \( i^3 = -i \), \( i^4 = 1 \), and then it repeats. Therefore, to simplify \( i^{26} \), we can find the remainder of 26 divided by 4 and use the cyclical pattern to determine the result.

The imaginary unit \( i \) is defined as \( i = \sqrt{-1} \). The powers of \( i \) follow a cyclical pattern:

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

This pattern repeats every four powers.

To simplify \( i^{26} \), we need to find the remainder when 26 is divided by 4. This is because the powers of \( i \) repeat every four terms.

\[ 26 \div 4 = 6 \text{ remainder } 2 \]

Thus, the remainder is 2.

Since the remainder is 2, we use the cyclical pattern of \( i \) to determine that:

\[ i^{26} = i^2 = -1 \]

Final Answer