simplifiy  i138 Show all work including steps to solve


To simplify the expression \( i^{138} \), where \( i \) is the imaginary unit, follow these steps:

### Step 1: Recall the powers of \( i \)
The powers of \( i \) repeat every four terms:
\[
\begin{aligned}
i^1 & = i, \\
i^2 & = -1, \\
i^3 & = -i, \\
i^4 & = 1.
\end{aligned}
\]
So, the cycle repeats every four powers.

### Step 2: Determine the remainder when 138 is divided by 4
To find which part of the cycle \( i^{138} \) corresponds to, divide 138 by 4 and find the remainder.

\[
138 \div 4 = 34 \text{ remainder } 2
\]

This means:
\[
138 = 4 \times 34 + 2
\]

So, \( i^{138} \) has the same value as \( i^2 \).

### Step 3: Use the power of \( i^2 \)
From the first step, we know:
\[
i^2 = -1
\]

### Final Answer:
Thus, \( i^{138} = -1 \).

Would you like more details or have any other questions? Here are some related questions:

1. How do you simplify \( i^{200} \)?
2. What are the properties of imaginary numbers?
3. How do you express complex numbers in polar form?
4. What is the geometric interpretation of multiplying by \( i \)?
5. How do powers of \( i \) relate to rotations in the complex plane?

**Tip:** When working with powers of \( i \), always reduce the exponent modulo 4 to simplify the expression.

simplify i^138. Show all work including steps to solve

Learn how to simplify i^138 by understanding the properties of the imaginary unit i. This detailed solution breaks down the steps, including the use of modulo operation and recognizing the repeating cycle of powers of i. Suitable for students in Grades 9-12.

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