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!