Question

One of the values of \[ (\sqrt{3}-i)^{5/3} \] is:

• A. \( 2^{5/3} \text{cis} \frac{5\pi}{18} \)
• B. \( 2^{5/3} \text{cis} \frac{19\pi}{18} \)
• C. \( 2^{5/3} \text{cis} \frac{23\pi}{18} \)
• D. \( 2^{5/3} \text{cis} \frac{17\pi}{18} \)

Hint:

When options are given in "cis" form, first find the basic argument. If it doesn't match, add \( 2\pi \) (or multiples) to the internal angle before multiplying by the power to find the branch that matches the choices.

Answer 1

The Correct Option is B

Concept: To find the fractional powers of a complex number, we use De Moivre's Theorem in its general form.
• General form: \( z = r \text{cis} (\theta + 2k\pi) \).
• Power: \( z^{p/q} = r^{p/q} \text{cis} \left( \frac{p(\theta + 2k\pi)}{q} \right) \).

Step 1: Expressing \( \sqrt{3} - i \) in polar form.
Magnitude \( r = \sqrt{(\sqrt{3})^2 + (-1)^2} = 2 \). Argument \( \theta = \tan^{-1}\left(\frac{-1}{\sqrt{3}}\right) = -\frac{\pi}{6} \). \[ \sqrt{3} - i = 2 \text{cis} \left( 2k\pi - \frac{\pi}{6} \right) = 2 \text{cis} \left( \frac{12k-1}{6} \pi \right) \]

Step 2: Applying the power \( 5/3 \).
\[ (\sqrt{3} - i)^{5/3} = 2^{5/3} \text{cis} \left[ \frac{5}{3} \left( \frac{12k-1}{6} \pi \right) \right] = 2^{5/3} \text{cis} \left( \frac{5(12k-1)\pi}{18} \right) \]

Step 3: Evaluating for different values of \( k \).
For \( k = 0 \): \( \text{cis} \left( -\frac{5\pi}{18} \right) \equiv \text{cis} \left( \frac{31\pi}{18} \right) \). For \( k = 1 \): \( \text{cis} \left( \frac{5 \times 11 \pi}{18} \right) = \text{cis} \left( \frac{55\pi}{18} \right) \). Simplifying \( \frac{55\pi}{18} \): \( \frac{55\pi}{18} = 2\pi + \frac{19\pi}{18} \equiv \frac{19\pi}{18} \). This matches Option (B).