Find the number of integral solutions of ${(1 - i)}^{n} = 2^{n}$

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Solution

$2^{n} = (\frac{1 - i}{\sqrt{2}}) .^{n} (\sqrt{2})^{n}$
$2^{n} = 2^{\frac{n}{2}} . (e^{i \frac{π}{4}})^{n}$
$2^{n} = 2^{\frac{n}{2}} . e^{i \frac{n π}{4}}$
$2^{n} . e^{i 2 n π} = 2^{\frac{n}{2}} . e^{i \frac{n π}{4}}$
Hence we get only one solution that is $n = 0$ .

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