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# Find the number of non - zero integral solutions of the equation $\;|1-i|^{x}=2^{x}$

$(a)\;1\qquad(b)\;0\qquad(c)\;2\qquad(d)\;4$

Explanation :
$\;|1-i|^{x}=2^{x}$
$(\sqrt{1^{2}+(-1)^{2}})^{x} = 2^{x}$
$(\sqrt{2})^{x} =2^{x}$
$2^{\large\frac{x}{2}} = 2^{x}$
$\large\frac{x}{2} =x$
$x=2x$
$2x-x=0$
$x=0$
Thus , 0 is the integral solution of the given equation . Therefore , the number of non - zero integral solutions of the equation is 0