Question

1) Rút gọn biểu thức:

P=\(\sqrt{2+\sqrt{3}}.\sqrt{2+\sqrt{2+\sqrt{3}}}.\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{3}}}}.\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{3}}}}\)

2) Cho x=\(\sqrt[3]{1+\sqrt{65}}-\sqrt[3]{\sqrt{65}-1}.\) Tính Q= x³+12x+2009

Answer 1

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

\(=\sqrt{2+\sqrt{3}}.\sqrt{2+\sqrt{2+\sqrt{3}}}.\sqrt{2-\sqrt{2+\sqrt{3}}}\)

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

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

b/ \(x=\sqrt[3]{1+\sqrt{65}}+\sqrt[3]{1-\sqrt{65}}\)

\(\Rightarrow x^3=2+3\sqrt[3]{1-65}.x\)

\(\Rightarrow x^3=2-12x\)

\(\Rightarrow x^3+12x=2\)

\(\Rightarrow Q=2+2009=2011\)