[tex] \frac{(2+\sqrt{3})^{2005}\cdot (2-\sqrt{3})^{2005}+1}{(2-\sqrt{3})^{2005}} \
\cdot\frac{[2(2-\sqrt{3}]^{2005}}{2^{2004}}=\\
= \frac{[(2+\sqrt{3})(2-\sqrt{3}){]^{2005}+1}}{(2-\sqrt{3})^{2005}} \cdot\
\frac{2^{2005}\cdot(2-\sqrt{3})^{2005}}{2^{2004}} =\\
= \frac{2^{2006}}{2^{2004}} =2^{2}=4[/tex]
