Question

Find the number of non-zero integral solution of the equation ${|1-i|}^x=2^x$

Answer 1

Given, ${|1-i|}^x=2^x$

$\Rightarrow {\left(\sqrt{1^2+{(-1)}^2}\right)}^x=2^x$..........$|z|=\sqrt{x^2+y^2}$

$\Rightarrow {\left(\sqrt{2}\right)}^x=2^x$

$\Rightarrow 2^{\frac{x}{2}}=2^x$

$\Rightarrow \frac{x}{2}=x$

$\Rightarrow x=2x$

$\Rightarrow x=0$

Thus, $x=0$ is the only integral solution of the given equation.

Therefore, the number of non-zero integral solutions of the given equation is $0$.