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Q.
If $(1-i)^n=2^n$, then $n$ is equal to
Complex Numbers and Quadratic Equations
Solution:
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$