Find the number of non-zero integral solutions of the equation $|1-i|^{x}=2^{x}$
Find the number of non-zero integral solutions of the equation $|1-i|^{x}=2^{x}$
$|1-i|^{x}=2^{x}$
$\Rightarrow(\sqrt{1^{2}+(-1)^{2}})^{x}=2^{x}$
$\Rightarrow(\sqrt{2})^{x}=2^{x}$
$\Rightarrow 2^{x / 2}=2^{x}$
$\Rightarrow \frac{x}{2}=x$
$\Rightarrow x=2 x$
$\Rightarrow 2 x-x=0$
$\Rightarrow x=0$
Thus, $0$ is the only integral solution of the given equation. Therefore, the number of nonzero integral solutions of the given equation is $0 .$
Similar Questions
Number of complex numbers $z$ such that $\left| z \right| + z - 3\bar z = 0$ is equal to
Number of complex numbers $z$ such that $\left| z \right| + z - 3\bar z = 0$ is equal to
Let $\bar{z}$ denote the complex conjugate of a complex number $z$. If $z$ is a non-zero complex number for which both real and imaginary parts of
$(\bar{z})^2+\frac{1}{z^2}$
are integers, then which of the following is/are possible value($s$) of $|z|$ ?
Let $\bar{z}$ denote the complex conjugate of a complex number $z$. If $z$ is a non-zero complex number for which both real and imaginary parts of
$(\bar{z})^2+\frac{1}{z^2}$
are integers, then which of the following is/are possible value($s$) of $|z|$ ?
- [IIT 2022]
If complex numbers $z_1$, $z_2$ are such that $\left| {{z_1}} \right| = \sqrt 2 ,\left| {{z_2}} \right| = \sqrt 3$ and $\left| {{z_1} + {z_2}} \right| = \sqrt {5 - 2\sqrt 3 }$, then the value of $|Arg z_1 -Arg z_2|$ is
If complex numbers $z_1$, $z_2$ are such that $\left| {{z_1}} \right| = \sqrt 2 ,\left| {{z_2}} \right| = \sqrt 3$ and $\left| {{z_1} + {z_2}} \right| = \sqrt {5 - 2\sqrt 3 }$, then the value of $|Arg z_1 -Arg z_2|$ is
If $(3 + i)z = (3 - i)\bar z,$then complex number $z$ is
If $(3 + i)z = (3 - i)\bar z,$then complex number $z$ is
Let $z$ be a complex number such that $\left| z \right| + z = 3 + i$ (where $i = \sqrt { - 1} $). Then $\left| z \right|$ is equal to
Let $z$ be a complex number such that $\left| z \right| + z = 3 + i$ (where $i = \sqrt { - 1} $). Then $\left| z \right|$ is equal to
- [JEE MAIN 2019]