Simplify : `i^(38)`

Answer

To simplify \( i^{38} \), we can follow these steps:

### Step 1: Understand the powers of \( i \)
We know that:
- \( i = \sqrt{-1} \)
- \( i^2 = -1 \)
- \( i^3 = i^2 \cdot i = -1 \cdot i = -i \)
- \( i^4 = i^2 \cdot i^2 = (-1) \cdot (-1) = 1 \)

### Step 2: Identify the pattern
From the above calculations, we can see that the powers of \( i \) repeat every four terms:
- \( i^1 = i \)
- \( i^2 = -1 \)
- \( i^3 = -i \)
- \( i^4 = 1 \)
- \( i^5 = i \) (and so on)

### Step 3: Reduce the exponent modulo 4
To simplify \( i^{38} \), we can reduce the exponent modulo 4:
\[ 38 \mod 4 \]
Calculating this gives:
\[ 38 \div 4 = 9 \quad \text{(remainder 2)} \]
So, \( 38 \mod 4 = 2 \).

### Step 4: Substitute back into the pattern
Now, we can substitute back:
\[ i^{38} = i^2 \]

### Step 5: Use the known value of \( i^2 \)
From our earlier calculations, we know:
\[ i^2 = -1 \]

### Final Answer
Thus, we conclude:
\[ i^{38} = -1 \]

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