Question

Calculati:$[(2+ \sqrt{3}) ^{2005} + \frac{1}{(2- \sqrt{3}) ^{2005} } ]$ · $\frac{ (4-2 \sqrt{3}) ^{2005} }{ 2^{2004} }$

Answer 1

[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]