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 \[ \boxed{8} \]

Answer 2

Given:
We are to find the product of all four values of:
\[ \left( 1 + i\sqrt{3} \right)^{3/4} \]

Step 1: Convert the complex number to polar form
Let \( z = 1 + i\sqrt{3} \)
Find modulus \( |z| \):
\[ |z| = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2 \]
Argument \( \theta = \arg(z) = \tan^{-1}\left( \frac{\sqrt{3}}{1} \right) = \frac{\pi}{3} \)

So, in polar form:
\[ z = 2 \text{cis} \frac{\pi}{3} = 2\left( \cos\frac{\pi}{3} + i\sin\frac{\pi}{3} \right) \]

Step 2: Raise to the power \( \frac{3}{4} \)
\[ z^{3/4} = \left( 2 \text{cis} \frac{\pi}{3} \right)^{3/4} = 2^{3/4} \text{cis} \left( \frac{3\pi}{4} + \frac{2\pi k}{4} \right), \quad k = 0, 1, 2, 3 \]
So the four values are:
\[ 2^{3/4} \text{cis} \left( \frac{3\pi}{4} \right), \quad 2^{3/4} \text{cis} \left( \frac{5\pi}{4} \right), \quad 2^{3/4} \text{cis} \left( \frac{7\pi}{4} \right), \quad 2^{3/4} \text{cis} \left( \frac{9\pi}{4} \right) \]

Step 3: Take the product of all values
Each value has modulus \( 2^{3/4} \), so the product of the four values is:
\[ \left( 2^{3/4} \right)^4 = 2^3 = 8 \]
The arguments are evenly spaced on the unit circle, so the imaginary parts cancel out in multiplication, and the argument sum doesn’t affect the modulus result.

Final Answer:
The product of the four values is 8.