\[\begin{array}{l} {\left( {\sqrt {4 - \sqrt {15} } } \right)^x} + {\left( {\sqrt {4 + \sqrt {15} } } \right)^x} = 8 \hfill \\ 4 + \sqrt {15} = \frac{{\left( {4 + \sqrt {15} } \right)\left( {4 - \sqrt {15} } \right)}}{{4 - \sqrt {15} }} = \frac{1}{{4 - \sqrt {15} }} \hfill \\ {\left( {\sqrt {4 - \sqrt {15} } } \right)^x} + \frac{1}{{{{\left( {\sqrt {4 - \sqrt {15} } } \right)}^x}}} = 8 \hfill \\ {\left( {\sqrt {4 - \sqrt {15} } } \right)^x} = t \hfill \\ t + \frac{1}{t} = 8 \Leftrightarrow \left[ \begin{array}{l} t = 4 + \sqrt {15} \hfill \\ t = 4 - \sqrt {15} \hfill \\ \end{array} \right. \Leftrightarrow \left[ \begin{array}{l} {\left( {\sqrt {4 - \sqrt {15} } } \right)^x} = 4 + \sqrt {15} \hfill \\ {\left( {\sqrt {4 - \sqrt {15} } } \right)^x} = 4 - \sqrt {15} \hfill \\ \end{array} \right. \Leftrightarrow \hfill \\ \left[ \begin{array}{l} {\left( {4 - \sqrt {15} } \right)^{\frac{x}{2}}} = {\left( {4 - \sqrt {15} } \right)^{ - 1}} \hfill \\ {\left( {4 - \sqrt {15} } \right)^{\frac{x}{2}}} = {\left( {4 - \sqrt {15} } \right)^1} \hfill \\ \end{array} \right. \Leftrightarrow \left[ \begin{array}{l} \frac{x}{2} = - 1 \hfill \\ \frac{x}{2} = 1 \hfill \\ \end{array} \right. \Leftrightarrow \left[ \begin{array}{l} x = - 2 \hfill \\ x = 2 \hfill \\ \end{array} \right. \hfill \\ \end{array}\]

\[ \pm 2\]