\(A=\sqrt{3+\sqrt{5+2\sqrt{3}}}.\sqrt{3-\sqrt{5+2\sqrt{3}}}\)
\(=\sqrt{\left(3+\sqrt{5+2\sqrt{3}}\right)\left(3-\sqrt{5+2\sqrt{3}}\right)}\)
\(=\sqrt{3^2-\left(5+2\sqrt{3}\right)}\)
\(=\sqrt{4-2\sqrt{3}}\)
\(=\sqrt{3-2\sqrt{3}+1}\)
\(=\sqrt{\left(\sqrt{3}-1\right)^2}=\left|\sqrt{3}-1\right|\)
\(=\sqrt{3}-1\)
\(B=\sqrt{4+\sqrt{8}}.\sqrt{2+\sqrt{2+\sqrt{2}}}.\sqrt{2-\sqrt{2+\sqrt{2}}}\)
\(=\sqrt{4+\sqrt{2^2.2}}.\sqrt{\left(2+\sqrt{2+\sqrt{2}}\right)\left(2-\sqrt{2+\sqrt{2}}\right)}\)
\(=\sqrt{4+2\sqrt{2}}.\sqrt{2^2-\left(2+\sqrt{2}\right)}\)
\(=\sqrt{2\left(2+\sqrt{2}\right)}.\sqrt{2-\sqrt{2}}\)
\(=\sqrt{2\left(2+\sqrt{2}\right)\left(2-\sqrt{2}\right)}\)
\(=\sqrt{2\left(4-2\right)}=\sqrt{4}=2\)
\(C=\left(\sqrt{12}+12\sqrt{15}-4\sqrt{135}\right).\sqrt{3}\)
\(=\left(\sqrt{12}+12\sqrt{15}-4\sqrt{3^2.15}\right).\sqrt{3}\)
\(=\left(\sqrt{12}+12\sqrt{15}-4.3\sqrt{15}\right).\sqrt{3}\)
\(=\left(\sqrt{12}+12\sqrt{15}-12\sqrt{15}\right).\sqrt{3}\)
\(=\sqrt{12}.\sqrt{3}=\sqrt{36}=6\)
\(D=2\sqrt{40\sqrt{12}}-2\sqrt{\sqrt{75}}-3\sqrt{5\sqrt{48}}\)
\(=2\sqrt{40.\sqrt{2^2.3}}-2\sqrt{\sqrt{5^2.3}}-3\sqrt{5.\sqrt{4^2.3}}\)
\(=2\sqrt{80\sqrt{3}}-2\sqrt{5\sqrt{3}}-3\sqrt{20\sqrt{3}}\)
\(=2\sqrt{4^2.5\sqrt{3}}-2\sqrt{5\sqrt{3}}-3\sqrt{2^2.5\sqrt{3}}\)
\(=8\sqrt{5\sqrt{3}}-2\sqrt{5\sqrt{3}}-6\sqrt{5\sqrt{3}}\)
\(=0\)
