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

$\begin{array}{l}(\sqrt{3}+1{)}^{2\mathrm{x}}+(\sqrt{3}-1{)}^{2\mathrm{x}}=2^{3\mathrm{x}}=(2\sqrt{2}{)}^{2\mathrm{x}}\\ \Rightarrow {\left(\frac{\sqrt{3}+1}{2\sqrt{2}}\right)}^{2\mathrm{x}}+{\left(\frac{\sqrt{3}-1}{2\sqrt{2}}\right)}^{2\mathrm{x}}=1\\ \Rightarrow {\left(\sin {75}^{\circ }\right)}^{2\mathrm{x}}+{\left(\cos {75}^{\circ }\right)}^{2\mathrm{x}}=1\Rightarrow \mathrm{x}=1\end{array}$