\(\sqrt{2+\sqrt{3}}\cdot\sqrt{2+\sqrt{2+\sqrt{3}}}\cdot\sqrt{2+\sqrt{2}+\sqrt{2+\sqrt{3}}}\cdot\sqrt{2-\sqrt{2+\sqrt{3}}}\)
\(=\sqrt{\left(2+\sqrt{3}\right)\cdot\left(2+\sqrt{2+\sqrt{3}}\right)\cdot\left(2+\sqrt{2}+\sqrt{2+\sqrt{3}}\right)\cdot\left(2-\sqrt{2+\sqrt{3}}\right)}\)
\(=\sqrt{\left(2+\sqrt{3}\right)\cdot\left(4-\left(2+\sqrt{3}\right)\right)\cdot\left(2+\sqrt{2}+\sqrt{2+\sqrt{3}}\right)}\)
\(=\sqrt{\left(2+\sqrt{3}\right)\cdot\left(4-2-\sqrt{3}\right)\cdot\left(2+\sqrt{2}+\sqrt{2+\sqrt{3}}\right)}\)
\(=\sqrt{\left(2+\sqrt{3}\right)\cdot\left(2-\sqrt{3}\right)\cdot\left(2+\sqrt{2}+\sqrt{2+\sqrt{3}}\right)}\)
\(=\sqrt{\left(4-3\right)\cdot\left(2+\sqrt{2}+\sqrt{2+\sqrt{3}}\right)}\)
\(=\sqrt{1\left(2+\sqrt{2}+\sqrt{2+\sqrt{3}}\right)}\)
\(=\sqrt{2+\sqrt{2}+\sqrt{2+\sqrt{3}}}\)
