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

Options

A. `1-isqrt(3)`
B. `-1+isqrt(3)`
C. `isqrt(3)`
D. `-isqrt(3)`

Explanation

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 first recognize that \( -\frac{1}{2} + \frac{i\sqrt{3}}{2} \) can be expressed in polar form.

### Step 1: Identify the complex number
The complex number \( z = -\frac{1}{2} + \frac{i\sqrt{3}}{2} \) can be written in polar form.

- The modulus \( r \) is given by:
\[ r = \sqrt{\left(-\frac{1}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2} = \sqrt{\frac{1}{4} + \frac{3}{4}} = \sqrt{1} = 1 \]

- The argument \( \theta \) is:
\[ \theta = \tan^{-1}\left(\frac{\frac{\sqrt{3}}{2}}{-\frac{1}{2}}\right) = \tan^{-1}(-\sqrt{3}) = \frac{2\pi}{3} \text{ (in the second quadrant)} \]

Thus, we can express \( z \) as:
\[ z = e^{i\frac{2\pi}{3}} \]

### Step 2: Calculate the powers of \( z \)
Now, we compute \( z^{334} \) and \( z^{365} \):
- Using De Moivre's theorem:
\[ z^{n} = r^n e^{in\theta} \]
Since \( r = 1 \):
\[ z^{334} = e^{i\frac{2\pi}{3} \cdot 334} = e^{i\frac{668\pi}{3}} = e^{i(222\pi + \frac{2\pi}{3})} = e^{i\frac{2\pi}{3}} = z \]
(Since \( e^{i2\pi k} = 1 \) for any integer \( k \))

For \( z^{365} \):
\[ z^{365} = e^{i\frac{2\pi}{3} \cdot 365} = e^{i\frac{730\pi}{3}} = e^{i(243\pi + \frac{1\pi}{3})} = e^{i\frac{\pi}{3}} = \frac{1}{2} + \frac{i\sqrt{3}}{2} \]

### Step 3: Substitute back into the expression
Now we substitute back into the original expression:
\[ 4 + 5z + 3\left(\frac{1}{2} + \frac{i\sqrt{3}}{2}\right) \]
Calculating \( 3\left(\frac{1}{2} + \frac{i\sqrt{3}}{2}\right) \):
\[ 3\left(\frac{1}{2} + \frac{i\sqrt{3}}{2}\right) = \frac{3}{2} + \frac{3i\sqrt{3}}{2} \]

Now we combine everything:
\[ 4 + 5\left(-\frac{1}{2} + \frac{i\sqrt{3}}{2}\right) + \left(\frac{3}{2} + \frac{3i\sqrt{3}}{2}\right) \]
Calculating \( 5z \):
\[ 5z = 5\left(-\frac{1}{2} + \frac{i\sqrt{3}}{2}\right) = -\frac{5}{2} + \frac{5i\sqrt{3}}{2} \]

### Step 4: Combine all parts
Now we combine:
\[ 4 - \frac{5}{2} + \frac{3}{2} + \left(\frac{5i\sqrt{3}}{2} + \frac{3i\sqrt{3}}{2}\right) \]
Calculating the real part:
\[ 4 - \frac{5}{2} + \frac{3}{2} = 4 - 1 = 3 \]
Calculating the imaginary part:
\[ \frac{5i\sqrt{3}}{2} + \frac{3i\sqrt{3}}{2} = \frac{8i\sqrt{3}}{2} = 4i\sqrt{3} \]

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