Question

1, So sánh \(\sqrt{1+\sqrt{2+\sqrt{3}}}\) và 2
2,Tính
     A=\(\sqrt{2+\sqrt{3}}\sqrt{2+\sqrt{2+\sqrt{3}}}\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{3}}}}\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{3}}}}\)
 

Answer 1

2, \(A=\sqrt{2+\sqrt{3}}\sqrt{2+\sqrt{2+\sqrt{3}}}\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{3}}}}\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{3}}}}\)

\(=\dfrac{\sqrt{4-3}}{\sqrt{2-\sqrt{3}}}.\dfrac{\sqrt{4-\left(2+\sqrt{3}\right)}}{\sqrt{2-\sqrt{2+\sqrt{3}}}}.\dfrac{\sqrt{4-\left(2+\sqrt{2+\sqrt{3}}\right)}}{\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{3}}}}}.\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{3}}}}\)

\(=\dfrac{\left(\sqrt{2-\sqrt{3}}\right)\left(\sqrt{2-\sqrt{2+\sqrt{3}}}\right)\left(\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{3}}}}\right)}{\left(\sqrt{2-\sqrt{3}}\right)\left(\sqrt{2-\sqrt{2+\sqrt{3}}}\right)\left(\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{3}}}}\right)}\)

\(=1\)