Question

Answer 1

\(=\sqrt{3+\sqrt{2}}\cdot\sqrt{3+\sqrt{6+\sqrt{2}}}\cdot\sqrt{9-6-\sqrt{6+\sqrt{2}}}\)

\(=\sqrt{3+\sqrt{2}}\cdot\sqrt{3+\sqrt{6+\sqrt{2}}}\cdot\sqrt{3-\sqrt{6+\sqrt{2}}}\)

\(=\sqrt{3+\sqrt{2}}\cdot\sqrt{9-6-\sqrt{2}}\)

\(=\sqrt{3+\sqrt{2}}\cdot\sqrt{3-\sqrt{2}}=\sqrt{7}\)

Answer 2

\(P=\sqrt{3+\sqrt{2}}.\sqrt{3+\sqrt{6+\sqrt{2}}}.\sqrt{3+\sqrt{6+\sqrt{6+\sqrt{2}}}}.\sqrt{3-\sqrt{6+\sqrt{6+\sqrt{2}}}}\)

\(=\sqrt{3+\sqrt{2}.}\sqrt{3+\sqrt{6+\sqrt{2}}}.\sqrt{\left(3+\sqrt{6+\sqrt{6+\sqrt{2}}}\right).\left(3-\sqrt{6+\sqrt{6+\sqrt{2}}}\right)}\)

\(=\sqrt{3+\sqrt{2}}.\sqrt{3+\sqrt{6+\sqrt{2}}}.\sqrt{9-\left(6+\sqrt{6+\sqrt{2}}\right)}\)

\(=\sqrt{3+\sqrt{2}}.\sqrt{3+\sqrt{6+\sqrt{2}}}.\sqrt{3-\sqrt{6+\sqrt{2}}}\)

\(=\sqrt{3+\sqrt{2}}.\sqrt{\left(3+\sqrt{6+\sqrt{2}}\right)\left(3-\sqrt{6+\sqrt{2}}\right)}=\sqrt{3+\sqrt{2}}.\sqrt{3-\sqrt{2}}\)

\(=\sqrt{\left(3+\sqrt{2}\right)\left(3-\sqrt{2}\right)}=\sqrt{5}\)