If ` ( sqrt(3) + i) ^(100) = 2^(99) (a + ib) , " then " a^(2) + b^(2)` is equal to

Options

A. `sqrt(2)`
B. 4
C. `sqrt(3)`
D. none of these

Answer

To solve the problem, we need to find \( a^2 + b^2 \) given that

\[ (\sqrt{3} + i)^{100} = 2^{99} (a + ib). \]

### Step 1: Convert \(\sqrt{3} + i\) to polar form

First, we find the modulus and argument of \(\sqrt{3} + i\).

- The modulus \( r \) is given by:

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

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

\[ \theta = \tan^{-1}\left(\frac{1}{\sqrt{3}}\right) = \frac{\pi}{6}. \]

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:

\[ (\sqrt{3} + i)^{100} = (2 e^{i \frac{\pi}{6}})^{100} = 2^{100} e^{i \frac{100\pi}{6}} = 2^{100} e^{i \frac{50\pi}{3}}. \]

### Step 3: Simplify the argument

Next, we simplify the argument \( \frac{50\pi}{3} \):

\[ \frac{50\pi}{3} = 16\pi + \frac{2\pi}{3} \quad (\text{since } 16\pi = 5 \cdot 2\pi + 6\pi). \]

Thus, we can reduce it to:

\[ e^{i \frac{50\pi}{3}} = e^{i \frac{2\pi}{3}}. \]

### Step 4: Rewrite the expression

Now we can rewrite our expression:

\[ (\sqrt{3} + i)^{100} = 2^{100} e^{i \frac{2\pi}{3}}. \]

### Step 5: Express in terms of \( a + ib \)

We know from the problem statement that:

\[ (\sqrt{3} + i)^{100} = 2^{99} (a + ib). \]

Equating both sides gives us:

\[ 2^{100} e^{i \frac{2\pi}{3}} = 2^{99} (a + ib). \]

Dividing both sides by \( 2^{99} \):

\[ 2 e^{i \frac{2\pi}{3}} = a + ib. \]

### Step 6: Find \( a \) and \( b \)

Now we can express \( a \) and \( b \):

\[ a + ib = 2 \left(\cos\frac{2\pi}{3} + i \sin\frac{2\pi}{3}\right). \]

Calculating the cosine and sine:

- \( \cos\frac{2\pi}{3} = -\frac{1}{2} \)
- \( \sin\frac{2\pi}{3} = \frac{\sqrt{3}}{2} \)

Thus,

\[ a = 2 \left(-\frac{1}{2}\right) = -1, \]
\[ b = 2 \left(\frac{\sqrt{3}}{2}\right) = \sqrt{3}. \]

### Step 7: Calculate \( a^2 + b^2 \)

Now we can find \( a^2 + b^2 \):

\[ a^2 + b^2 = (-1)^2 + (\sqrt{3})^2 = 1 + 3 = 4. \]

### Final Answer

Thus, the value of \( a^2 + b^2 \) is

\[ \boxed{4}. \]

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