Question

If $(1-i{)}^n=2^n$, then $n$ is equal to

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

Answer 1

Answer: (B)

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