IF `(sqrt3 + i)^10 = a + i b`, then `a` and `b` are respectively

Options

A. 128 & `128sqrt(3)`
B. 64 and `64sqrt(3)`
C. 512 and `512sqrt(3)`
D. none of these

Answer

To solve the problem \( (\sqrt{3} + i)^{10} = a + ib \), we will follow these steps:

### Step 1: Convert to Polar Form
First, we need to express \( \sqrt{3} + i \) in polar form. The modulus \( r \) is calculated as follows:
\[ r = \sqrt{(\sqrt{3})^2 + (1)^2} = \sqrt{3 + 1} = \sqrt{4} = 2 \]
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:
\[ \sqrt{3} + i = 2 \left( \cos\frac{\pi}{6} + i \sin\frac{\pi}{6} \right) \]

### Step 2: Raise to the Power of 10
Now, we raise this expression to the power of 10:
\[ (\sqrt{3} + i)^{10} = \left( 2 \left( \cos\frac{\pi}{6} + i \sin\frac{\pi}{6} \right) \right)^{10} \]
Using De Moivre's theorem, we have:
\[ = 2^{10} \left( \cos\left(10 \cdot \frac{\pi}{6}\right) + i \sin\left(10 \cdot \frac{\pi}{6}\right) \right) \]
Calculating \( 2^{10} \):
\[ 2^{10} = 1024 \]
Now, we simplify the angle:
\[ 10 \cdot \frac{\pi}{6} = \frac{10\pi}{6} = \frac{5\pi}{3} \]

### Step 3: Evaluate Cosine and Sine
Next, we find \( \cos\frac{5\pi}{3} \) and \( \sin\frac{5\pi}{3} \):
\[ \cos\frac{5\pi}{3} = \cos\left(2\pi - \frac{\pi}{3}\right) = \cos\frac{\pi}{3} = \frac{1}{2} \]
\[ \sin\frac{5\pi}{3} = \sin\left(2\pi - \frac{\pi}{3}\right) = -\sin\frac{\pi}{3} = -\frac{\sqrt{3}}{2} \]

### Step 4: Substitute Back
Now we substitute these values back into our expression:
\[ (\sqrt{3} + i)^{10} = 1024 \left( \frac{1}{2} - i \frac{\sqrt{3}}{2} \right) \]
Distributing \( 1024 \):
\[ = 1024 \cdot \frac{1}{2} - 1024 \cdot i \cdot \frac{\sqrt{3}}{2} \]
\[ = 512 - 512\sqrt{3} i \]

### Step 5: Identify \( a \) and \( b \)
From the expression \( 512 - 512\sqrt{3} i \), we can identify:
\[ a = 512, \quad b = -512\sqrt{3} \]

### Final Answer
Thus, the values of \( a \) and \( b \) are:
\[ \boxed{512} \text{ and } \boxed{-512\sqrt{3}} \]