Simplify: `i^(18)`

Answer

To simplify \( i^{18} \), we can follow these steps:

### Step 1: Understand the powers of \( i \)
We know the following:
- \( i^1 = i \)
- \( i^2 = -1 \)
- \( i^3 = -i \)
- \( i^4 = 1 \)

### Step 2: Reduce the exponent modulo 4
Since the powers of \( i \) repeat every 4, we can reduce the exponent \( 18 \) modulo \( 4 \):
\[ 18 \mod 4 = 2 \]
This means \( i^{18} \) is equivalent to \( i^2 \).

### Step 3: Substitute the reduced exponent
Now we can substitute \( i^2 \):
\[ i^{18} = i^2 \]

### Step 4: Use the known value of \( i^2 \)
From our earlier knowledge:
\[ i^2 = -1 \]

### Final Answer
Thus, we have:
\[ i^{18} = -1 \]