If `i = sqrt-1 ` then `4 + 5 ( - (1)/(2) + (i sqrt3)/( 2)) ^( 334) + 3 (- (1)/(2) + ( i sqrt3)/( 2))^365` is equal to to

Options

A. `1- i sqrt3`
B. `-1 + sqrt3`
C. `i sqrt3`
D. `-i sqrt3`

Answer

To solve the expression \( 4 + 5 \left( -\frac{1}{2} + \frac{i \sqrt{3}}{2} \right)^{334} + 3 \left( -\frac{1}{2} + \frac{i \sqrt{3}}{2} \right)^{365} \), we will use the concept of cube roots of unity.

### Step-by-Step Solution:

1. **Identify the complex number**:
Let \( \omega = -\frac{1}{2} + \frac{i \sqrt{3}}{2} \). This is one of the cube roots of unity, specifically \( \omega = e^{2\pi i / 3} \).

2. **Properties of cube roots of unity**:
The cube roots of unity satisfy the equation \( \omega^3 = 1 \) and \( 1 + \omega + \omega^2 = 0 \).

3. **Calculate powers of \( \omega \)**:
- Since \( \omega^3 = 1 \), we can reduce the exponents modulo 3:
- \( 334 \mod 3 = 1 \) (since \( 334 = 3 \times 111 + 1 \))
- \( 365 \mod 3 = 2 \) (since \( 365 = 3 \times 121 + 2 \))

4. **Substitute back into the expression**:
\[ 4 + 5 \omega^{334} + 3 \omega^{365} = 4 + 5 \omega^1 + 3 \omega^2 \]
This simplifies to:
\[ 4 + 5 \omega + 3 \omega^2 \]

5. **Use the identity \( 1 + \omega + \omega^2 = 0 \)**:
From this identity, we can express \( \omega^2 \) in terms of \( \omega \):
\[ \omega^2 = -1 - \omega \]
Substitute this into the expression:
\[ 4 + 5 \omega + 3(-1 - \omega) = 4 + 5 \omega - 3 - 3 \omega \]
This simplifies to:
\[ 4 - 3 + (5 - 3) \omega = 1 + 2 \omega \]

6. **Substituting the value of \( \omega \)**:
Now substitute \( \omega = -\frac{1}{2} + \frac{i \sqrt{3}}{2} \):
\[ 1 + 2\left(-\frac{1}{2} + \frac{i \sqrt{3}}{2}\right) = 1 - 1 + i \sqrt{3} = i \sqrt{3} \]

### Final Result:
Thus, the value of the expression is:
\[ \boxed{i \sqrt{3}} \]