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

Simplify the expression. i^26 i^26=
Transcript text: Simplify the expression. \[ \begin{array}{l} i^{26} \\ i^{26}= \end{array} \]
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Solution

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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.

Step 1: Determine the Cyclical Pattern of \( i \)

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.

Step 2: Calculate the Remainder

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.

Step 3: Use the Remainder to Simplify the Expression

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

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

Final Answer

\(\boxed{-1}\)

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