Given `z=(1+isqrt(3))^(100),` then `[R E(z)//I M(z)]` equals `2^(100)` b. `2^(50)` c. `1/(sqrt(3))` d. `sqrt(3)`

Options

A. `2^(100)`
B. `2^(50)`
C. `(1)/(sqrt(3))`
D. `sqrt(3)`

Explanation

To solve the problem, we need to find the value of \(\frac{Re(z)}{Im(z)}\) where \(z = (1 + i\sqrt{3})^{100}\).

### Step 1: Convert \(1 + i\sqrt{3}\) to polar form
The complex number \(1 + i\sqrt{3}\) can be expressed in polar form as follows:
- The modulus \(r\) is given by:
\[ r = |1 + i\sqrt{3}| = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2 \]
- 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:
\[ 1 + i\sqrt{3} = 2\left(\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}\right) \]

### Step 2: Raise to the power of 100
Now we raise \(1 + i\sqrt{3}\) to the power of 100:
\[ z = (1 + i\sqrt{3})^{100} = \left(2\left(\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}\right)\right)^{100} \]
Using De Moivre's theorem:
\[ z = 2^{100}\left(\cos\left(100 \cdot \frac{\pi}{3}\right) + i\sin\left(100 \cdot \frac{\pi}{3}\right)\right) \]

### Step 3: Simplify the angle
Now we simplify \(100 \cdot \frac{\pi}{3}\):
\[ 100 \cdot \frac{\pi}{3} = \frac{100\pi}{3} \]
To find the equivalent angle within \(0\) to \(2\pi\), we can subtract \(2\pi\) multiples:
\[ \frac{100\pi}{3} - 2\pi \cdot 16 = \frac{100\pi}{3} - \frac{96\pi}{3} = \frac{4\pi}{3} \]
Thus:
\[ z = 2^{100}\left(\cos\frac{4\pi}{3} + i\sin\frac{4\pi}{3}\right) \]

### Step 4: Calculate \(\cos\frac{4\pi}{3}\) and \(\sin\frac{4\pi}{3}\)
From the unit circle:
- \(\cos\frac{4\pi}{3} = -\frac{1}{2}\)
- \(\sin\frac{4\pi}{3} = -\frac{\sqrt{3}}{2}\)

### Step 5: Substitute back into \(z\)
Now substituting these values back into \(z\):
\[ z = 2^{100}\left(-\frac{1}{2} + i\left(-\frac{\sqrt{3}}{2}\right)\right) = 2^{100}\left(-\frac{1}{2} - i\frac{\sqrt{3}}{2}\right) \]
This simplifies to:
\[ z = 2^{100} \cdot -\frac{1}{2} - i \cdot 2^{100} \cdot \frac{\sqrt{3}}{2} \]
Thus:
\[ z = -2^{99} - i\sqrt{3} \cdot 2^{99} \]

### Step 6: Identify the real and imaginary parts
From the expression for \(z\):
- \(Re(z) = -2^{99}\)
- \(Im(z) = -\sqrt{3} \cdot 2^{99}\)

### Step 7: Calculate \(\frac{Re(z)}{Im(z)}\)
Now we compute:
\[ \frac{Re(z)}{Im(z)} = \frac{-2^{99}}{-\sqrt{3} \cdot 2^{99}} = \frac{2^{99}}{\sqrt{3} \cdot 2^{99}} = \frac{1}{\sqrt{3}} \]

### Final Answer
Thus, the value of \(\frac{Re(z)}{Im(z)}\) is:
\[ \frac{1}{\sqrt{3}} \]