` (4+3i) +(7-4i)-(3+5i)+i^(25)` is equal to

Options

A. 2+7i
B. 4+7i
C. ` 8-5i`
D. `-8+2i`

Answer

To solve the expression \( (4 + 3i) + (7 - 4i) - (3 + 5i) + i^{25} \), we will follow these steps:

### Step 1: Simplify \( i^{25} \)
We know that \( i^4 = 1 \). Therefore, we can express \( i^{25} \) in terms of powers of \( i \):
\[ i^{25} = i^{24} \cdot i = (i^4)^6 \cdot i = 1^6 \cdot i = i \]

### Step 2: Substitute \( i^{25} \) back into the expression
Now we can substitute \( i^{25} \) with \( i \):
\[ (4 + 3i) + (7 - 4i) - (3 + 5i) + i \]

### Step 3: Combine the real parts and the imaginary parts
Now we will group the real and imaginary parts together:
- Real parts: \( 4 + 7 - 3 \)
- Imaginary parts: \( 3i - 4i - 5i + i \)

Calculating the real parts:
\[ 4 + 7 - 3 = 8 \]

Calculating the imaginary parts:
\[ 3i - 4i - 5i + i = (3 - 4 - 5 + 1)i = -5i \]

### Step 4: Combine the results
Now we can combine the results of the real and imaginary parts:
\[ 8 - 5i \]

### Final Answer
Thus, the expression \( (4 + 3i) + (7 - 4i) - (3 + 5i) + i^{25} \) simplifies to:
\[ \boxed{8 - 5i} \]

---