[tex] \frac{5+ \sqrt{5} }{8- \sqrt{5} } =
\frac{5+ \sqrt{5} }{8- \sqrt{5} } * \frac{8+\sqrt{5}}{8+\sqrt{5}} =
\frac{(5+\sqrt{5})*(8+\sqrt{5})}{(8-\sqrt{5})*(8+\sqrt{5})} =
\frac{40+5 \sqrt{5}+8 \sqrt{5}+( \sqrt{5})^2 }{64+8\sqrt{5}-8\sqrt{5}-( \sqrt{5})^2} =
[/tex]
[tex] \frac{40+13 \sqrt{5}+5 }{64-5} =
\boxed{\frac{45+13 \sqrt{5} }{59} }[/tex]
