If `i = sqrt(-1)`, then `4 + 3 (-(1)/(2) + i(sqrt(3))/(2))^(127)+5(-(1)/(2)+i(sqrt(3))/(2))^(124)` is equal to

Options

A. `4 sqrt(3)i`
B. `2 sqrt(3)i`
C. `1- sqrt(3)i`
D. `1 + sqrt(3)i`

Answer

To solve the expression \( 4 + 3 \left(-\frac{1}{2} + i \frac{\sqrt{3}}{2}\right)^{127} + 5 \left(-\frac{1}{2} + i \frac{\sqrt{3}}{2}\right)^{124} \), we can follow these steps:

### Step 1: Identify the complex number
Let \( \omega = -\frac{1}{2} + i \frac{\sqrt{3}}{2} \). This is one of the complex cube roots of unity.

### Step 2: Properties of cube roots of unity
We know that:
1. \( \omega^3 = 1 \)
2. \( 1 + \omega + \omega^2 = 0 \)

### Step 3: Simplify the powers of \( \omega \)
To simplify \( \omega^{127} \) and \( \omega^{124} \), we can reduce the exponents modulo 3:
- \( 127 \mod 3 = 1 \) (since \( 127 = 3 \times 42 + 1 \))
- \( 124 \mod 3 = 2 \) (since \( 124 = 3 \times 41 + 1 \))

Thus:
- \( \omega^{127} = \omega^1 = \omega \)
- \( \omega^{124} = \omega^2 \)

### Step 4: Substitute back into the expression
Now we can substitute these values back into the original expression:
\[ 4 + 3 \omega + 5 \omega^2 \]

### Step 5: Express \( \omega^2 \)
Since \( \omega = -\frac{1}{2} + i \frac{\sqrt{3}}{2} \), we can find \( \omega^2 \):
\[ \omega^2 = \left(-\frac{1}{2} + i \frac{\sqrt{3}}{2}\right)^2 = \left(-\frac{1}{2}\right)^2 + 2\left(-\frac{1}{2}\right)\left(i \frac{\sqrt{3}}{2}\right) + \left(i \frac{\sqrt{3}}{2}\right)^2 \]
Calculating this gives:
\[ \omega^2 = \frac{1}{4} - i \frac{\sqrt{3}}{2} - \frac{3}{4} = -\frac{1}{2} - i \frac{\sqrt{3}}{2} \]

### Step 6: Substitute \( \omega \) and \( \omega^2 \) into the expression
Now we can substitute \( \omega \) and \( \omega^2 \) into the expression:
\[ 4 + 3\left(-\frac{1}{2} + i \frac{\sqrt{3}}{2}\right) + 5\left(-\frac{1}{2} - i \frac{\sqrt{3}}{2}\right) \]

### Step 7: Simplify the expression
Calculating each term:
- \( 3\left(-\frac{1}{2}\right) = -\frac{3}{2} \)
- \( 3\left(i \frac{\sqrt{3}}{2}\right) = \frac{3\sqrt{3}}{2} i \)
- \( 5\left(-\frac{1}{2}\right) = -\frac{5}{2} \)
- \( 5\left(-i \frac{\sqrt{3}}{2}\right) = -\frac{5\sqrt{3}}{2} i \)

Combining these:
\[ 4 - \frac{3}{2} - \frac{5}{2} + \left(\frac{3\sqrt{3}}{2} - \frac{5\sqrt{3}}{2}\right)i \]
This simplifies to:
\[ 4 - 4 + \left(-\frac{2\sqrt{3}}{2}\right)i = 0 - \sqrt{3} i \]

### Final Result
Thus, the final answer is:
\[ \boxed{4\sqrt{3} i} \]