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$$\sqrt{2+\sqrt{3}}\times\sqrt{2+\sqrt{2+\sqrt{3}}}\times\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{3}}}}\times\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{3}}}}$$ is equal to
Let, $$\sqrt{\ 2+\sqrt{\ 3}}=x$$
So, the given expression, $$\sqrt{2+\sqrt{3}}\times\sqrt{2+\sqrt{2+\sqrt{3}}}\times\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{3}}}}\times\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{3}}}}$$ can be written as:
=$$x\sqrt{\ 2+x}\sqrt{\ 2+\sqrt{\ 2+x}}\sqrt{\ 2-\sqrt{\ 2+x}}$$
=$$x\sqrt{\ 2+x}\sqrt{\ \left(2\right)^2-\left(\sqrt{\ 2+x}\right)^2}$$
=$$x\sqrt{\ 2+x}\sqrt{\ 4-\left(2+x\right)}$$
=$$x\sqrt{\ 2+x}\sqrt{\ 2-x}$$
=$$x\sqrt{\ 2^2-x^2}$$
=$$x\sqrt{4-x^2}$$
=$$x\sqrt{4-\left(\sqrt{\ 2+\sqrt{\ 3}}\right)^2}$$
=$$x\sqrt{4-\left(\ 2+\sqrt{\ 3}\right)}$$
=$$x\sqrt{2-\sqrt{\ 3}}$$
=$$\sqrt{\ 2+\sqrt{\ 3}}\sqrt{2-\sqrt{\ 3}}$$
=$$\sqrt{\left(2\right)^2-\left(\sqrt{\ 3}\right)^2}$$
=$$\sqrt{\ 4-3}$$
=1