Question

Value of $x$, satisfying ${(\sqrt{3}+1)}^{2x}+{(\sqrt{3}-1)}^{2x}=2^{3x}$, is equal to

Answer 1

Given,

$(\sqrt{3}+1{)}^{2x}+(\sqrt{3}-1{)}^{2x}=2^{3x}$

$\Rightarrow {\left\{\right(\sqrt{3}+1{)}^2\}}^x+{\left\{\right(\sqrt{3}-1{)}^2\}}^x=2^{3x}$

$\Rightarrow (4+2\sqrt{3}{)}^x+(4-2\sqrt{3}{)}^x=2^{3x}$

$\Rightarrow 2^x(2+\sqrt{3}{)}^x+2^x(2-\sqrt{3}{)}^x=2^{3x}$

$\Rightarrow (2+\sqrt{3}{)}^x+(2-\sqrt{3}{)}^x=2^{2x}$

Clearly given equation satisfied only for $x=1$

$\therefore$ Number of value of $x$ is $1$