[tex] \it ( \frac{ \sqrt{3} - 1}{ \sqrt{2} } - \frac{ \sqrt{2} }{ \sqrt{3} + 1 } )^{15} = \\ \\ \it ( \frac{( \sqrt{3} - 1) \sqrt{2} }{2} - \frac{ \sqrt{2} \times ( \sqrt{3} - 1)}{2})^{15 } = \\ \\ \it ( \frac{ \sqrt{6} - \sqrt{2} }{2} - \frac{ \sqrt{6} - \sqrt{2} }{2} )^{15} = \\ \\ \it \: {0}^{15} = \\ \\ \boxed{\it 0} [/tex]