1: \(\left(\sqrt{5}+\sqrt{3}\right)\cdot\sqrt{8-2\sqrt{15}}\)
\(=\left(\sqrt{5}+\sqrt{3}\right)\cdot\sqrt{5-2\cdot\sqrt{5}\cdot\sqrt{3}+3}\)
\(=\left(\sqrt{5}+\sqrt[]{3}\right)\cdot\sqrt{\left(\sqrt{5}-\sqrt{3}\right)^2}\)
\(=\left(\sqrt{5}+\sqrt{3}\right)\left(\sqrt{5}-\sqrt{3}\right)\)
=5-3=2
2: \(\left(\sqrt{10}+\sqrt[]{6}\right)\cdot\sqrt{8-2\sqrt{15}}\)
\(=\left(\sqrt{5}+\sqrt{3}\right)\cdot\sqrt{2}\cdot\sqrt{\left(\sqrt{5}-\sqrt{3}\right)^2}\)
\(=\sqrt{2}\left(\sqrt{5}+\sqrt{3}\right)\left(\sqrt{5}-\sqrt{3}\right)\)
\(=\sqrt{2}\left(5-3\right)=2\sqrt{2}\)
3: \(\left(5+2\sqrt{3}\right)\cdot\sqrt{37-20\sqrt{3}}\)
\(=\left(5+2\sqrt{3}\right)\cdot\sqrt{25-2\cdot5\cdot2\sqrt{3}+12}\)
\(=\left(5+2\sqrt{3}\right)\sqrt{\left(5-2\sqrt{3}\right)^2}\)
\(=\left(5+2\sqrt{3}\right)\left(5-2\sqrt{3}\right)=25-12=13\)
4: \(\left(2+\sqrt{3}\right)\cdot\sqrt{7-4\sqrt{3}}\)
\(=\left(2+\sqrt{3}\right)\cdot\sqrt{\left(2-\sqrt{3}\right)^2}\)
\(=\left(2+\sqrt{3}\right)\left(2-\sqrt{3}\right)=4-3=1\)
5: \(\sqrt{2-\sqrt{3}}\left(\sqrt{6}+\sqrt{2}\right)\)
\(=\sqrt{4-2\sqrt{3}}\left(\sqrt{3}+1\right)\)
\(=\left(\sqrt{3}+1\right)\cdot\sqrt{\left(\sqrt{3}-1\right)^2}\)
\(=\left(\sqrt{3}+1\right)\left(\sqrt{3}-1\right)=3-1=2\)
6: \(\left(\sqrt{10}-\sqrt{14}\right)\cdot\sqrt{6+\sqrt{35}}\)
\(=\left(\sqrt{5}-\sqrt{7}\right)\cdot\sqrt{12+2\sqrt{35}}\)
\(=\left(\sqrt{5}-\sqrt{7}\right)\cdot\left(\sqrt{5}+\sqrt{7}\right)\)
=5-7=-2
7: \(\left(\sqrt{6}+\sqrt{10}\right)\cdot\sqrt{4-\sqrt{15}}\)
\(=\left(\sqrt{5}+\sqrt{3}\right)\sqrt{8-2\sqrt{15}}\)
\(=\left(\sqrt{5}+\sqrt{3}\right)\cdot\sqrt{\left(\sqrt{5}-\sqrt{3}\right)^2}\)
\(=\left(\sqrt{5}+\sqrt{3}\right)\left(\sqrt{5}-\sqrt{3}\right)=5-3=2\)
8: \(\sqrt{2+\sqrt{9+4\sqrt{2}}}\)
\(=\sqrt{2+\sqrt{8+2\cdot2\sqrt{2}\cdot1+1}}\)
\(=\sqrt{2+\sqrt{\left(2\sqrt{2}+1\right)^2}}=\sqrt{2+2\sqrt{2}+1}\)
\(=\sqrt{\left(\sqrt{2}+1\right)^2}=\sqrt{2}+1\)
9: \(\sqrt{5-\sqrt{13+\sqrt{48}}}\)
\(=\sqrt{5-\sqrt{12+2\cdot2\sqrt{3}\cdot1+1}}\)
\(=\sqrt{5-\sqrt{\left(2\sqrt{3}+1\right)^2}}=\sqrt{4-2\sqrt{3}}\)
\(=\sqrt{\left(\sqrt{3}-1\right)^2}=\sqrt{3}-1\)
10: \(\sqrt{8+2\sqrt{6-\sqrt{20}}}\)
\(=\sqrt{8+2\sqrt{5-2\sqrt{5}+1}}\)
\(=\sqrt{8+2\sqrt{\left(\sqrt{5}-1\right)^2}}\)
\(=\sqrt{8+2\left(\sqrt{5}-1\right)}=\sqrt{6+2\sqrt{5}}\)
\(=\sqrt{5+2\cdot\sqrt{5}\cdot1+1}=\sqrt{\left(\sqrt{5}+1\right)^2}=\sqrt{5}+1\)
11: \(\sqrt{\sqrt{5}-\sqrt{6-2\sqrt{5}}}\)
\(=\sqrt{\sqrt{5}-\sqrt{\left(\sqrt{5}-1\right)^2}}\)
\(=\sqrt{\sqrt{5}-\left(\sqrt{5}-1\right)}=\sqrt{\sqrt{5}-\sqrt{5}+1}\)
\(=\sqrt{1}=1\)
12: \(\sqrt{3-\sqrt{29-12\sqrt{5}}}\)
\(=\sqrt{3-\sqrt{20-2\cdot2\sqrt{5}\cdot3+9}}\)
\(=\sqrt{3-\sqrt{\left(2\sqrt{5}-3\right)^2}}\)
\(=\sqrt{3-\left(2\sqrt{5}-3\right)}=\sqrt{3-2\sqrt{5}+3}\)
\(=\sqrt{5-2\cdot\sqrt{5}\cdot1+1}=\sqrt{\left(\sqrt{5}-1\right)^2}=\sqrt{5}-1\)