1: \(A=\sqrt{2}\cdot\left(\sqrt{2+\sqrt{3}}+\sqrt{2-\sqrt{3}}\right)\)
\(=\sqrt{4+2\sqrt{3}}+\sqrt{4-2\sqrt{3}}\)
\(=\sqrt{\left(\sqrt{3}+1\right)^2}+\sqrt{\left(\sqrt{3}-1\right)^2}\)
\(=\sqrt{3}+1+\sqrt{3}-1=2\sqrt{3}\)
2: \(B=\sqrt{\dfrac{8-\sqrt{15}}{2}}+\sqrt{\dfrac{8+\sqrt{15}}{2}}\)
\(=\sqrt{\dfrac{16-2\sqrt{15}}{4}}+\sqrt{\dfrac{16+2\sqrt{15}}{4}}\)
\(=\dfrac{\sqrt{\left(\sqrt{15}-1\right)^2}}{2}+\dfrac{\sqrt{\left(\sqrt{15}+1\right)^2}}{2}\)
\(=\dfrac{\sqrt{15}-1+\sqrt{15}+1}{2}=\dfrac{2\sqrt{15}}{2}=\sqrt{15}\)
3: \(C=\left(5+\sqrt{21}\right)\left(\sqrt{14}-\sqrt{6}\right)\cdot\sqrt{5-\sqrt{21}}\)
\(=\left(5+\sqrt{21}\right)\left(\sqrt{7}-\sqrt{3}\right)\cdot\sqrt{10-2\sqrt{21}}\)
\(=\left(5+\sqrt{21}\right)\left(\sqrt{7}-\sqrt{3}\right)\left(\sqrt{7}-\sqrt{3}\right)\)
\(=\left(5+\sqrt{21}\right)\left(10-2\sqrt{21}\right)\)
\(=50-10\sqrt{21}+10\sqrt{21}-42=8\)
4: \(D=\dfrac{\sqrt{2-\sqrt{3}}}{\sqrt{2}}=\dfrac{\sqrt{4-2\sqrt{3}}}{2}\)
\(=\dfrac{\sqrt{\left(\sqrt{3}-1\right)^2}}{2}=\dfrac{\sqrt{3}-1}{2}\)
5: \(E=\left(4+\sqrt{15}\right)\left(\sqrt{10}-\sqrt{6}\right)\cdot\sqrt{4-\sqrt{15}}\)
\(=\left(4+\sqrt{15}\right)\left(\sqrt{5}-\sqrt{3}\right)\cdot\sqrt{8-2\sqrt{15}}\)
\(=\left(4+\sqrt{15}\right)\left(\sqrt{5}-\sqrt{3}\right)\cdot\left(\sqrt{5}-\sqrt{3}\right)\)
\(=\left(4+\sqrt{15}\right)\left(8-2\sqrt{15}\right)\)
\(=2\left(4+\sqrt{15}\right)\left(4-\sqrt{15}\right)=2\left(16-15\right)=2\)
6: \(F=\sqrt{3-\sqrt{5}}\left(\sqrt{10}-\sqrt{2}\right)\cdot\left(3+\sqrt{5}\right)\)
\(=\sqrt{6-2\sqrt{5}}\left(\sqrt{5}-1\right)\left(3+\sqrt{5}\right)\)
\(=\left(\sqrt{5}-1\right)\cdot\left(\sqrt{5}-1\right)\left(3+\sqrt{5}\right)\)
\(=\left(6-2\sqrt{5}\right)\left(3+\sqrt{5}\right)=2\left(3-\sqrt{5}\right)\cdot\left(3+\sqrt{5}\right)\)
\(=2\left(9-5\right)=2\cdot4=8\)
