Question

The product of the four values of \( \left( 1 + i\sqrt{3} \right)^{3/4} \) is:

• A. \( -8i \)
• B. \( i \)
• C. \( -8 \)
• D. \( 8 \)

Hint:

The product of all distinct roots of a complex number raised to a power is simply the magnitude raised to the power of the number of roots.

Answer 1

The Correct Option is D

We express \( 1 + i\sqrt{3} \) in polar form: \[ 1 + i\sqrt{3} = 2 \left( \cos \frac{\pi}{3} + i \sin \frac{\pi}{3} \right) \] Using De Moivre's Theorem, we get: \[ \left( 1 + i\sqrt{3} \right)^{3/4} = 2^{3/4} \left( \cos \frac{3\pi}{4} + i \sin \frac{3\pi}{4} \right) \] The product of the four values is the magnitude raised to the power 4: \[ (2^{3/4})^4 = 2^3 = 8 \] % Final Answer \[ \boxed{8} \]