Find the number of non-zero integral solution | vedclass

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Question

(A) Given equation: $|1-i|^{x} = 2^{x}$ Calculate the modulus: $|1-i| = \sqrt{1^{2} + (-1)^{2}} = \sqrt{2}$ Substitute the modulus into the equation:

Vedclass · Accepted Answer

$0$

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(A) Given equation: $|1-i|^{x} = 2^{x}$ Calculate the modulus: $|1-i| = \sqrt{1^{2} + (-1)^{2}} = \sqrt{2}$ Substitute the modulus into the equation: $(\sqrt{2})^{x} = 2^{x}$ Rewrite $\sqrt{2}$ as $2^{1/2}$: $(2^{1/2})^{x} = 2^{x}$ Simplify the exponents: $2^{x/2} = 2^{x}$ Equate the exponents: $\frac{x}{2} = x$ Multiply by $2$: $x = 2x$ Rearrange: $2x - x = 0 \Rightarrow x = 0$ The only integral solution is $x = 0$. Since $0$ is not a non-zero integer,the number of non-zero integral solutions is $0$.

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

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