Question

$\sqrt{2} * \sqrt{2+ \sqrt{2}} * {\sqrt{2 + \sqrt{2+ \sqrt{2}} } * {\sqrt{2 - \sqrt{2+ \sqrt{2} }}$

Answer 1

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$\sqrt{2} * \sqrt{2+ \sqrt{2} } * \sqrt{(2+ \sqrt{2+ \sqrt{2} })(2- \sqrt{2+ \sqrt{2} }) } = \\ \\ \sqrt{2} * \sqrt{2+ \sqrt{2} } * \sqrt{2^2-( \sqrt{2+ \sqrt{2} })^2 } } = \\ \\ \sqrt{2} * \sqrt{2+ \sqrt{2} } * \sqrt{4-(2+ \sqrt{2}) } = \\ \\ \sqrt{2} * \sqrt{2+ \sqrt{2} } * \sqrt{4-2- \sqrt{2} } = \\ \\ \sqrt{2} * \sqrt{2+ \sqrt{2} } * \sqrt{2- \sqrt{2} } = \\ \\ \sqrt{2} * \sqrt{(2+ \sqrt{2})(2- \sqrt{2}) } = \\ \\ \sqrt{2} * \sqrt{2^2-( \sqrt{2})^2 } =$

$\sqrt{2} * \sqrt{4-2} = \\ \\ \sqrt{2} * \sqrt{2} = \\ \\ \sqrt{4} =\fbox{ 2 }$