Write the following in the form x+iy:
`i^(9)+i^(10)+i^(11)+i^(12)`.

Answer

To solve the expression \( i^{9} + i^{10} + i^{11} + i^{12} \) and write it in the form \( x + iy \), we can follow these steps:

### Step 1: Understand the powers of \( i \)
The powers of \( i \) (the imaginary unit) cycle every four terms:
- \( i^{1} = i \)
- \( i^{2} = -1 \)
- \( i^{3} = -i \)
- \( i^{4} = 1 \)

### Step 2: Calculate each power
Now, let's calculate \( i^{9} \), \( i^{10} \), \( i^{11} \), and \( i^{12} \) using the cyclic property:
- \( i^{9} = i^{(8+1)} = i^{8} \cdot i^{1} = 1 \cdot i = i \)
- \( i^{10} = i^{(8+2)} = i^{8} \cdot i^{2} = 1 \cdot (-1) = -1 \)
- \( i^{11} = i^{(8+3)} = i^{8} \cdot i^{3} = 1 \cdot (-i) = -i \)
- \( i^{12} = i^{(8+4)} = i^{8} \cdot i^{4} = 1 \cdot 1 = 1 \)

### Step 3: Substitute the values back into the expression
Now we can substitute these values back into the original expression:
\[ i^{9} + i^{10} + i^{11} + i^{12} = i + (-1) + (-i) + 1 \]

### Step 4: Simplify the expression
Combine the real parts and the imaginary parts:
- Real parts: \( -1 + 1 = 0 \)
- Imaginary parts: \( i - i = 0 \)

Thus, we have:
\[ i^{9} + i^{10} + i^{11} + i^{12} = 0 + 0i \]

### Step 5: Write in the form \( x + iy \)
Finally, we can express the result in the form \( x + iy \):
\[ 0 + 0i \]

### Final Answer:
The expression \( i^{9} + i^{10} + i^{11} + i^{12} \) in the form \( x + iy \) is \( 0 + 0i \).

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