Question

If $(1-i{)}^n=2^n,thenn=$
[RPET 1990]

• A. 1
• B. 0
• C. -1
• D. None of these

Answer 1

The correct option is B

0

If $(1-i{)}^n=2^n$ ......(i)
We know that if two complex numbers are equal, their moduli must also be equal, therefore from (i), we have
$|(1-i{)}^n|=|2^n|\Rightarrow |1-i{|}^n=|2{|}^n,(\because 2^n>0)$
$\Rightarrow \left[\sqrt{1^2+(-1{)}^2}\right]=2^n\Rightarrow (\sqrt{2}{)}^n=2^n$
$2^{n/2}=2^n\Rightarrow \frac{n}{2}=n\Rightarrow n=0$

Trick : By inspection, $(1-i{)}^0=2^0\Rightarrow 1=1$