Question

calculati $[(2+ \sqrt{3} )^{2015} + \frac{1}{(2- \sqrt{3} )^{2015}}]* \frac{(4-2 \sqrt{3})^{2015} }{2^{2014}}$

Answer 1

   
[tex]\displaystyle \left[(2+ \sqrt{3} )^{2015}+\frac{1}{(2-\sqrt{3} )^{2015}}\right] \times \frac{(4-2\sqrt{3})^{2015}}{2^{2014}}= \\ \\ =\left[ \frac{(2+ \sqrt{3})^{2015}}{1} +\frac{1}{(2-\sqrt{3} )^{2015}}\right] \times \frac{[2(2-\sqrt{3})]^{2015}}{2^{2014}}= \\ \\ =\left[ \frac{(2+ \sqrt{3})^{2015} \times (2-\sqrt{3} )^{2015}}{(2-\sqrt{3} )^{2015}} +\frac{1}{(2-\sqrt{3} )^{2015}}\right] \times \frac{[2(2-\sqrt{3})]^{2015}}{2^{2014}}= [/tex]


[tex]\displaystyle =\left[ \frac{[(2+ \sqrt{3}) \times (2-\sqrt{3} )]^{2015}}{(2-\sqrt{3} )^{2015}} +\frac{1}{(2-\sqrt{3} )^{2015}}\right] \times \frac{[2(2-\sqrt{3})]^{2015}}{2^{2014}}= \\ \\ =\left[\frac{[4-3 ]^{2015}}{(2-\sqrt{3} )^{2015}} +\frac{1}{(2-\sqrt{3}) ^{2015}}\right] \times \frac{[2(2-\sqrt{3})]^{2015}}{2^{2014}}= \\ \\ =\left[\frac{1}{(2-\sqrt{3} )^{2015}} +\frac{1}{(2-\sqrt{3}) ^{2015}}\right] \times \frac{[2(2-\sqrt{3})]^{2015}}{2^{2014}}= [/tex]


[tex]\displaystyle =\frac{2}{(2-\sqrt{3}) ^{2015}} \times \frac{2^{2015} \times (2-\sqrt{3})^{2015}}{2^{2014}}= \\ \\ = \frac{2 \times 2^{2015} \times (2-\sqrt{3})^{2015}}{ (2-\sqrt{3}) ^{2015} \times 2^{2014}}= \\ \\ =\frac{2 \times 2^{2015} }{ 2^{2014}}= \frac{2^{2015+1} }{ 2^{2014}}= \frac{2^{2016} }{ 2^{2014}}= 2^{2016-2014}=2^2 = \boxed{4} [/tex]