Question

All values of \( (8i)^{\frac{1}{3}} \) are:

• A. \( \pm (\sqrt{3} + i), -2i \)
• B. \( \pm \sqrt{3} + i, -2i \)
• C. \( \pm (\sqrt{3} - i), 2i \)
• D. \( \pm (2 + i), i \)

Hint:

For complex roots, express the number in polar form and use De Moivre’s theorem.

Answer 1

The Correct Option is B

Step 1: Convert to Polar Form Expressing \( 8i \) in polar form: \[ 8i = 8 \text{cis} \frac{\pi}{2} \]
Step 2: Using De Moivre’s Theorem The cube roots are given by: \[ z_k = 8^{\frac{1}{3}} \text{cis} \left( \frac{\frac{\pi}{2} + 2k\pi}{3} \right), \quad k = 0,1,2 \] Calculating values, we get: \[ \boxed{\pm (\sqrt{3} + i), -2i} \]