Express the following in the form a + ib
`i^9`

Answer

To express \( i^9 \) in the form \( a + ib \), we can follow these steps:

### Step 1: Understand the powers of \( i \)
We know that \( i \) is defined as \( \sqrt{-1} \) and has the following powers:
- \( i^1 = i \)
- \( i^2 = -1 \)
- \( i^3 = -i \)
- \( i^4 = 1 \)

After \( i^4 \), the powers of \( i \) repeat every four terms.

### Step 2: Reduce the exponent modulo 4
To simplify \( i^9 \), we can reduce the exponent modulo 4:
\[ 9 \mod 4 = 1 \]
This means:
\[ i^9 = i^{4 \cdot 2 + 1} = (i^4)^2 \cdot i^1 = 1^2 \cdot i = i \]

### Step 3: Write \( i \) in the form \( a + ib \)
Now we can express \( i \) in the form \( a + ib \):
\[ i = 0 + 1i \]
Thus, we have \( a = 0 \) and \( b = 1 \).

### Final Answer
Therefore, \( i^9 \) expressed in the form \( a + ib \) is:
\[ 0 + 1i \]