Question

\[ (1+i\sqrt{3})^{10}+(1-i\sqrt{3})^{10} \] equals to:

• A. \(1024\)
• B. \(-1024\)
• C. \(-624\)
• D. \(624\)

Hint:

Use De Moivre's theorem for powers of complex numbers in polar form.

Answer 1

The Correct Option is B

Concept:
Convert complex numbers into polar form and then use De Moivre's theorem.

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

Step 2: Raise to power \(10\). \[ (1+i\sqrt{3})^{10}=2^{10}\left(\cos\frac{10\pi}{3}+i\sin\frac{10\pi}{3}\right) \] \[ (1-i\sqrt{3})^{10}=2^{10}\left(\cos\frac{10\pi}{3}-i\sin\frac{10\pi}{3}\right) \]

Step 3: Add both. \[ (1+i\sqrt{3})^{10}+(1-i\sqrt{3})^{10} =2\cdot 2^{10}\cos\frac{10\pi}{3} \] \[ =2048\cos\frac{10\pi}{3} \] Now: \[ \cos\frac{10\pi}{3}=\cos\frac{4\pi}{3}=-\frac{1}{2} \] \[ =2048\left(-\frac{1}{2}\right) \] \[ =-1024 \] \[ \therefore \text{Correct Answer is (B)} \]