If `( sqrt(3) + i) ^(100) = 2 ^(99)` (a + i b) then b =

Options

A. `sqrt(3)`
B. `sqrt(2)`
C. 1
D. none of these

Answer

To solve the problem, we need to find the value of \( b \) in the expression \( ( \sqrt{3} + i)^{100} = 2^{99} (a + ib) \).

### Step 1: Convert \( \sqrt{3} + i \) to polar form
First, we need to express \( \sqrt{3} + i \) in polar form. The modulus \( r \) is given by:
\[ r = \sqrt{(\sqrt{3})^2 + (1)^2} = \sqrt{3 + 1} = \sqrt{4} = 2 \]
Next, we find the argument \( \theta \):
\[ \theta = \tan^{-1}\left(\frac{1}{\sqrt{3}}\right) = \frac{\pi}{6} \]
Thus, we can express \( \sqrt{3} + i \) in polar form as:
\[ \sqrt{3} + i = 2 \left( \cos\frac{\pi}{6} + i \sin\frac{\pi}{6} \right) = 2 e^{i \frac{\pi}{6}} \]

### Step 2: Raise to the power of 100
Now, we raise this expression to the power of 100:
\[ (\sqrt{3} + i)^{100} = \left(2 e^{i \frac{\pi}{6}}\right)^{100} = 2^{100} e^{i \frac{100\pi}{6}} = 2^{100} e^{i \frac{50\pi}{3}} \]

### Step 3: Simplify the argument
Next, we simplify the argument \( \frac{50\pi}{3} \):
\[ \frac{50\pi}{3} = 16\pi + \frac{2\pi}{3} \quad (\text{since } 16\pi \text{ is a multiple of } 2\pi) \]
Thus, we can write:
\[ e^{i \frac{50\pi}{3}} = e^{i \frac{2\pi}{3}} \]

### Step 4: Write in rectangular form
Now we can express \( e^{i \frac{2\pi}{3}} \) in rectangular form:
\[ e^{i \frac{2\pi}{3}} = \cos\frac{2\pi}{3} + i \sin\frac{2\pi}{3} = -\frac{1}{2} + i \frac{\sqrt{3}}{2} \]

### Step 5: Combine results
Now we combine the results:
\[ (\sqrt{3} + i)^{100} = 2^{100} \left(-\frac{1}{2} + i \frac{\sqrt{3}}{2}\right) = 2^{100} \left(-\frac{1}{2}\right) + 2^{100} \left(i \frac{\sqrt{3}}{2}\right) \]
This simplifies to:
\[ -2^{99} + i 2^{99} \sqrt{3} \]

### Step 6: Compare with given expression
We have:
\[ -2^{99} + i 2^{99} \sqrt{3} = 2^{99} (a + ib) \]
From this, we can equate the real and imaginary parts:
- Real part: \( a = -1 \)
- Imaginary part: \( b = \sqrt{3} \)

### Conclusion
Thus, the value of \( b \) is:
\[ \boxed{\sqrt{3}} \]