Question

calculati $(1+ \sqrt{2+ \sqrt{3}}) (1- \sqrt{2+ \sqrt{3}})-(1+ \sqrt{2- \sqrt{3}}) (1- \sqrt{2- \sqrt{3}})=$

Answer 1

$(1+ \sqrt{2+ \sqrt{3}}) (1- \sqrt{2+ \sqrt{3}})-(1+ \sqrt{2- \sqrt{3}}) (1- \sqrt{2- \sqrt{3}})= \\ = -(\sqrt{2+ \sqrt{3}}+1) (\sqrt{2+ \sqrt{3}}-1)-(1+ \sqrt{2- \sqrt{3}}) (1- \sqrt{2- \sqrt{3}})= \\ = -((\sqrt{2+ \sqrt{3}})^{2}-(1)^2)-((1)^2- \sqrt{2- \sqrt{3}})^{2})= \\ =-((2+ \sqrt{3})-1)-(1- (2 -\sqrt{3})) = \\ =-(2+ \sqrt{3}-1)-(1- 2 +\sqrt{3})= \\= -2- \sqrt{3}+1-1+ 2 -\sqrt{3}= \\ = -2+2 +1-1 -\sqrt{3}-\sqrt{3}= \\ = -2\sqrt{3}$