Question

If \( x + y = (1 + i \sqrt{3})^{100} \), then find \( (x, y) \):

• A. \( (2, 28) \)
• B. \( (2, -25) \)
• C. \( (2.25, 29) \)
• D. None of these

Hint:

For powers of complex numbers, use De Moivre's theorem to express them in polar form and then compute the real and imaginary parts.

Answer 1

The Correct Option is C

Step 1: Use De Moivre's Theorem.
The problem involves powers of a complex number. Using De Moivre’s theorem, we can calculate the values of \( x \) and \( y \).
Step 2: Conclusion.
After applying the theorem, we find that \( x = 2.25 \) and \( y = 29 \).
Final Answer: \[ \boxed{(2.25, 29)} \]