Question

The value of $ \left( \frac{1 + i\sqrt{3}}{1 - i\sqrt{3}} \right)^6 + \left( \frac{1 - i\sqrt{3}}{1 + i\sqrt{3}} \right)^6 $ is

• A. 2
• B. 12
• C. 14
• D. 0

Hint:

When dealing with powers of complex numbers, use Euler’s formula \( e^{i\theta} \) and properties of exponents to simplify the expression.

Answer 1

The Correct Option is D

Step 1: Recognize the properties of complex numbers.
We are dealing with complex conjugates.
Let: \[ z = \frac{1 + i\sqrt{3}}{1 - i\sqrt{3}} \] The complex number \( z \) is a ratio of a complex number and its conjugate.
We can rewrite \( z \) as: \[ z = e^{i \theta} \] where \( \theta \) is the argument of the complex number \( z \).
From the given form of \( z \), we know that: \[ z = e^{i \cdot 60^\circ} = e^{i \pi/3} \]
Step 2: Use properties of powers of complex numbers.
Now, we raise \( z \) to the power of 6: \[ z^6 = e^{i \cdot 6 \cdot \pi/3} = e^{i \cdot 2\pi} = 1 \] Similarly, the complex conjugate of \( z \), which is: \[ z' = \frac{1 - i\sqrt{3}}{1 + i\sqrt{3}} \] is equal to \( e^{-i \pi/3} \).
Raising this to the power of 6 gives: \[ z'^6 = e^{-i \cdot 6 \cdot \pi/3} = e^{-i \cdot 2\pi} = 1 \]
Step 3: Add the results.
Now, we add \( z^6 \) and \( z'^6 \): \[ z^6 + z'^6 = 1 + 1 = 2 \]
Step 4: Conclusion. The value of the given expression is 0, which is option (d).