Question

The product of all the values of \(\bigl(\sqrt{3} - i\bigr)^{25}\) is ?

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

Hint:

For any non-zero complex number \(w\), the product of all \(n\) distinct \(n\)th roots of \(w\) equals \(w\).
- Always carefully expand and simplify the base \((\sqrt{3} - i)^2\) before deducing the product of roots.

Answer 1

The Correct Option is C

Step 1: Interpret the multi-valued expression.
\(\bigl(\sqrt{3} - i\bigr)^{\tfrac{2}{5}}\) refers to all 5 distinct values of the fifth-root of \(\bigl(\sqrt{3} - i\bigr)^2\). If we let \[ W \;=\; \bigl(\sqrt{3} - i\bigr)^2, \] then the 5 values of \(W^{1/5}\) multiply to \(W\). Step 2: Compute \(\bigl(\sqrt{3} - i\bigr)^2\).
Expand: \[ (\sqrt{3} - i)^2 = 3 - 2\sqrt{3}\,i + i^2 = 3 - 2\sqrt{3}\,i - 1 = 2 - 2\sqrt{3}\,i = 2\,(1 - \sqrt{3}\,i). \] Thus \[ W \;=\; 2\,(1 - \sqrt{3}\,i). \] Step 3: Product of the 5 distinct 5th roots of \(W\).
A fundamental result for complex \(n\)th roots shows that the product of all 5 distinct roots of \(W\) is precisely \(W\). Hence the product of all values of \(\bigl(\sqrt{3} - i\bigr)^{\tfrac{2}{5}}\) is \[ \boxed{2\,(1 - \sqrt{3}\,i)}. \]