Question

Sa se rezolve: A = $\sqrt{2} * \sqrt{2 + \sqrt{2} } * \sqrt{2 + \sqrt{2 + \sqrt{2} } } * \sqrt{2 - \sqrt{2 + \sqrt{2} } }$

Answer 1

   
[tex] \sqrt{2} \times \sqrt{2 + \sqrt{2}} \times \sqrt{2+\sqrt{2+\sqrt{2}}} \times \sqrt{2-\sqrt{2+\sqrt{2}}} = \\ \\ =\sqrt{2} \times \sqrt{2 + \sqrt{2}} \times \sqrt{\left(2+\sqrt{2+\sqrt{2}}\right) \times \left(2-\sqrt{2+\sqrt{2}}\right)}= \\ \\ =\sqrt{2} \times \sqrt{2 + \sqrt{2}} \times \sqrt{2^2-\left(\sqrt{2+\sqrt{2}}\right)^2}= \\ \\ =\sqrt{2} \times \sqrt{2 + \sqrt{2}} \times \sqrt{4-\left(2+\sqrt{2}\right)}= \\ \\ =\sqrt{2} \times \sqrt{2 + \sqrt{2}} \times \sqrt{4-2-\sqrt{2}}= [/tex]


[tex]=\sqrt{2} \times \sqrt{2 + \sqrt{2}} \times \sqrt{2-\sqrt{2}}= \\ \\ =\sqrt{2} \times \sqrt{(2 + \sqrt{2}) \times (2-\sqrt{2})}= \\ \\ =\sqrt{2} \times \sqrt{2^2 - (\sqrt{2})^2 }= \\ \\ =\sqrt{2} \times \sqrt{4 - 2 }= \sqrt{2} \times \sqrt{2 }= \boxed{2}[/tex]