`x+iy=(1-isqrt3)^100` , then `(x,y)=`

Options

A. `(2^(99),2^(99)sqrt(3))`
B. `(2^(99),-2^(99)sqrt(3))`
C. `(-2^(99),2^(99)sqrt(3))`
D. none of these

Explanation

To solve the problem \( x + iy = (1 - i\sqrt{3})^{100} \), we will follow these steps:

### Step 1: Convert the complex number to polar form
First, we need to express \( 1 - i\sqrt{3} \) in polar form. The modulus \( r \) and argument \( \theta \) can be calculated as follows:

\[ r = |1 - i\sqrt{3}| = \sqrt{1^2 + (-\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2 \]

The argument \( \theta \) is given by:

\[ \theta = \tan^{-1}\left(\frac{-\sqrt{3}}{1}\right) = \tan^{-1}(-\sqrt{3}) = -\frac{\pi}{3} \]

Thus, we can write:

\[ 1 - i\sqrt{3} = 2 \left(\cos\left(-\frac{\pi}{3}\right) + i\sin\left(-\frac{\pi}{3}\right)\right) \]

### Step 2: Apply De Moivre's Theorem
Now we can raise this to the power of 100 using De Moivre's Theorem:

\[ (1 - i\sqrt{3})^{100} = \left(2 \left(\cos\left(-\frac{\pi}{3}\right) + i\sin\left(-\frac{\pi}{3}\right)\right)\right)^{100} \]

This becomes:

\[ = 2^{100} \left(\cos\left(-\frac{100\pi}{3}\right) + i\sin\left(-\frac{100\pi}{3}\right)\right) \]

### Step 3: Simplify the angle
Next, we need to simplify the angle \( -\frac{100\pi}{3} \):

\[ -\frac{100\pi}{3} = -33\pi + \frac{\pi}{3} = -33\pi + \frac{\pi}{3} \quad (\text{since } -33\pi \text{ is a multiple of } 2\pi) \]

Thus, we can reduce it to:

\[ -\frac{100\pi}{3} \equiv \frac{\pi}{3} \quad (\text{mod } 2\pi) \]

### Step 4: Calculate cosine and sine
Now we can find the cosine and sine values:

\[ \cos\left(\frac{\pi}{3}\right) = \frac{1}{2}, \quad \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \]

### Step 5: Substitute back
Substituting these values back, we have:

\[ (1 - i\sqrt{3})^{100} = 2^{100} \left(\frac{1}{2} + i\frac{\sqrt{3}}{2}\right) \]

This simplifies to:

\[ = 2^{100} \cdot \frac{1}{2} + i \cdot 2^{100} \cdot \frac{\sqrt{3}}{2} = 2^{99} + i \cdot 2^{99}\sqrt{3} \]

### Step 6: Identify \( x \) and \( y \)
From the expression \( x + iy = 2^{99} + i \cdot 2^{99}\sqrt{3} \), we can identify:

\[ x = 2^{99}, \quad y = 2^{99}\sqrt{3} \]

### Final Result
Thus, the final answer is:

\[ (x, y) = (2^{99}, 2^{99}\sqrt{3}) \]