Simplify:
`i^(23)`

Answer

To simplify \( i^{23} \), we can follow these steps:

### Step 1: Express the exponent in terms of multiples of 4
We know that the powers of \( i \) (where \( i = \sqrt{-1} \)) repeat every 4 cycles:
- \( i^1 = i \)
- \( i^2 = -1 \)
- \( i^3 = -i \)
- \( i^4 = 1 \)

To simplify \( i^{23} \), we can express 23 in terms of a multiple of 4:
\[ 23 = 4 \times 5 + 3 \]

### Step 2: Rewrite the expression using the exponent
Now we can rewrite \( i^{23} \) as:
\[ i^{23} = i^{4 \times 5 + 3} = (i^4)^5 \cdot i^3 \]

### Step 3: Simplify using the known values of powers of \( i \)
Since \( i^4 = 1 \), we can substitute this into our expression:
\[ (i^4)^5 = 1^5 = 1 \]
Thus, we have:
\[ i^{23} = 1 \cdot i^3 = i^3 \]

### Step 4: Find the value of \( i^3 \)
From our earlier list of powers of \( i \):
\[ i^3 = -i \]

### Final Result
Therefore, the simplified form of \( i^{23} \) is:
\[ \boxed{-i} \]

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