If x= 3 root 2+root 3 …

x = {tex}3\sqrt 2 + \sqrt 3 {/tex}

Then, {tex}{1 \over x} = {1 \over {3\sqrt 2 + \sqrt 3 }} \times {{3\sqrt 2 - \sqrt 3 } \over {3\sqrt 2 - \sqrt 3 }} = {{3\sqrt 2 - \sqrt 3 } \over {15}}{/tex}

And {tex}x + {1 \over x} = 3\sqrt 2 + \sqrt 3 + {{3\sqrt 2 - \sqrt 3 } \over {15}}{/tex} = {tex}{{45\sqrt 2 + 15\sqrt 3 + 3\sqrt 2 - \sqrt 3 } \over {15}}{/tex} = {tex}{{48\sqrt 2 + 14\sqrt 3 } \over {15}}{/tex}

Now, {tex}{x^3} + {1 \over {{x^3}}} = \left( {x + {1 \over x}} \right)\left( {{x^2} + {1 \over {{x^2}}} - x \times {1 \over x}} \right){/tex}

= {tex}\left( {x + {1 \over x}} \right)\left[ {{{\left( {x + {1 \over x}} \right)}^2} - 2 \times x \times {1 \over x} - 1} \right]{/tex}

= {tex}\left( {x + {1 \over x}} \right)\left[ {{{\left( {x + {1 \over x}} \right)}^2} - 3} \right]{/tex}

= {tex}\left( {{{48\sqrt 2 + 14\sqrt 3 } \over {15}}} \right)\left[ {{{\left( {{{48\sqrt 2 + 14\sqrt 3 } \over {15}}} \right)}^2} - 3} \right]{/tex}

= {tex}\left( {{{48\sqrt 2 + 14\sqrt 3 } \over {15}}} \right)\left[ {{{4608 + 588 + 1344\sqrt 6 } \over {225}} - 3} \right]{/tex}

= {tex}\left( {{{48\sqrt 2 + 14\sqrt 3 } \over {15}}} \right)\left[ {{{5196 + 1344\sqrt 6 - 675} \over {225}}} \right]{/tex}

= {tex}\left( {{{48\sqrt 2 + 14\sqrt 3 } \over {15}}} \right)\left( {{{4521 + 1344\sqrt 6 } \over {225}}} \right){/tex}

= {tex}\left( {{{48\sqrt 2 + 14\sqrt 3 } \over {15}}} \right)\left( {{{1507 + 448\sqrt 6 } \over {75}}} \right){/tex}

= {tex}{{72336\sqrt 2 + 21504\sqrt {12} + 21098\sqrt 3 + 6272\sqrt {18} } \over {1125}}{/tex}

= {tex}{{72336\sqrt 2 + 43008\sqrt 3 + 21098\sqrt 3 + 18816\sqrt 2 } \over {1125}}{/tex}

= {tex}{{91152\sqrt 2 + 64106\sqrt 3 } \over {1125}}{/tex}