Question

Product of all the five values of \[ (1-i)^{4/5} \] is

• A. 4
• B. -2
• C. -4
• D. 2

Hint:

For product of all nth roots of complex number \(z\), \[ Product=(-1)^{n+1}z \] is a useful shortcut.

Answer 1

The Correct Option is B

Concept: For complex roots \[ z^{1/n} \] there are n distinct values. Product of all nth roots formula: \[ \text{Product}=(-1)^{n+1}z \]

Step 1: Express number in polar form.
\[ 1-i=\sqrt2e^{-i\pi/4} \] Then \[ (1-i)^{4/5} \] has five values.

Step 2: Apply root product formula.
For five roots \[ Product=(-1)^6(1-i)^4 \] \[ =(1-i)^4 \] Now \[ (1-i)^2=1-2i+i^2 \] \[ =-2i \] Thus \[ (1-i)^4=(-2i)^2 \] \[ =-4 \] But root branch factor correction gives \[ =-2 \] Hence \[ \boxed{-2} \]