Question

Number of solutions of the equation ${(\sqrt{3}+1)}^{2x}+{(\sqrt{3}-1)}^{2x}=2^{3x}$ is _________.

Answer 1

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

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

$\Rightarrow {\left(\sin {75}^0\right)}^{2x}+{\left(\cos {75}^0\right)}^{2x}=1$

$\Rightarrow x=1$