simplify i138 Show all work including steps to solve in a list
 make it simple


To simplify \( i^{138} \), follow these steps:

### Step 1: Understand the powers of \( i \)
The imaginary unit \( i \) follows a cyclic pattern for powers:

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

This pattern repeats every 4 powers, so we can reduce large exponents by dividing by 4.

### Step 2: Find the remainder when 138 is divided by 4
To reduce \( i^{138} \), divide 138 by 4 and find the remainder:

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

So, \( i^{138} \) is equivalent to \( i^2 \).

### Step 3: Use the value of \( i^2 \)
From the cyclic pattern:
\[
i^2 = -1
\]

### Conclusion:
\[
i^{138} = -1
\]

Thus, the simplified value of \( i^{138} \) is \( -1 \).

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Would you like more details or further explanations on any step?

Here are 5 related questions:
1. What is the value of \( i^{50} \)?
2. How can powers of \( i \) be used in solving complex number problems?
3. Why does the cycle of powers of \( i \) repeat every 4 steps?
4. What is \( i^{100} \) simplified?
5. How do we apply powers of \( i \) in polar form?

**Tip**: When simplifying powers of \( i \), always divide the exponent by 4 to find the remainder!

simplify i138 Show all work including steps to solve in a list make it simple

Discover how to simplify i^138 by understanding the cyclic pattern of powers of i. This detailed step-by-step solution explains how to divide the exponent and use the properties of i, making it suitable for students in Grades 9-12.

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