Question

One of the values of \[ \left( \frac{1 + i}{\sqrt{2}} \right)^{2/3} \] is:

• A. \( \frac{1}{2} \left( \sqrt{3} + i \right) \)
• B. \( -i \)
• C. \( i \)
• D. \( -\sqrt{3} + i \)

Hint:

To calculate powers and roots of complex numbers, express them in polar form and apply De Moivre’s Theorem.

Answer 1

The Correct Option is A

Step 1: We need to compute \( \left( \frac{1 + i}{\sqrt{2}} \right)^{2/3} \). First, write \( 1 + i \) in polar form as \( \sqrt{2} \left( \cos \frac{\pi}{4} + i \sin \frac{\pi}{4} \right) \).
Step 2: Using De Moivre’s Theorem, we get: \[ \left( \frac{1 + i}{\sqrt{2}} \right)^{2/3} = \frac{1}{2} \left( \sqrt{3} + i \right). \]
Final Answer: \[ \boxed{\frac{1}{2} \left( \sqrt{3} + i \right)} \]