Question

doar acel subpunct ...........

Answer 1

$\it\dfrac{(4-2\sqrt3)^{2012}}{2^{2011}}=\dfrac{[2(2-\sqrt3)]^{2012}}{2^{2011}}=\dfrac{2^{2012}(2-\sqrt3)^{2012}}{2^{2011}}=2(2-\sqrt3)^{2012}$

Expresia din enunț devine:

$\it E= (2+\sqrt3)^{2012}\cdot2(2-\sqrt3)^{2012}+\dfrac{1}{(2-\sqrt3)^{2012}}\cdot2(2-\sqrt3)^{2012}=\\ \\ =2[(2+\sqrt3)(2-\sqrt3)]^{2012}+2=2[2^2-(\sqrt3)^2]^{2012}+2=2\cdot(4-3)^{2012}+2=\\ \\ =2\cdot1^{2012}+2=2\cdot1+2=2+2=4$