1: \(A=\sqrt{14+6\sqrt{5}}-\sqrt{8-2\sqrt{15}}\)
\(=\sqrt{9+2\cdot3\cdot\sqrt{5}+5}-\sqrt{5-2\cdot\sqrt{5}\cdot\sqrt{3}+3}\)
\(=\sqrt{\left(3+\sqrt{5}\right)^2}-\sqrt{\left(\sqrt{5}-\sqrt{3}\right)^2}\)
\(=3+\sqrt{5}-\sqrt{5}+\sqrt{3}=3+\sqrt{3}\)
2: \(A=\sqrt{7-2\sqrt{10}}-\sqrt{7+2\sqrt{10}}\)
\(=\sqrt{5-2\cdot\sqrt{5}\cdot\sqrt{2}+2}-\sqrt{5+2\cdot\sqrt{5}\cdot\sqrt{2}+2}\)
\(=\sqrt{\left(\sqrt{5}-\sqrt{2}\right)^2}-\sqrt{\left(\sqrt{5}+\sqrt{2}\right)^2}\)
\(=\sqrt{5}-\sqrt{2}-\sqrt{5}-\sqrt{2}=-2\sqrt{2}\)
3: \(A=\sqrt{9+4\sqrt{2}}-\sqrt{9-4\sqrt{2}}\)
\(=\sqrt{8+2\cdot2\sqrt{2}\cdot1+1}-\sqrt{8-2\cdot2\sqrt{2}\cdot1+1}\)
\(=\sqrt{\left(2\sqrt{2}+1\right)^2}-\sqrt{\left(2\sqrt{2}-1\right)^2}\)
\(=2\sqrt{2}+1-2\sqrt{2}+1=2\)
4: \(A=\sqrt{5-2\sqrt{6}}+\sqrt{3+2\sqrt{2}}\)
\(=\sqrt{3-2\cdot\sqrt{3}\cdot\sqrt{2}+2}+\sqrt{2+2\cdot\sqrt{2}\cdot1+1}\)
\(=\sqrt{\left(\sqrt{3}-\sqrt{2}\right)^2}+\sqrt{\left(\sqrt{2}+1\right)^2}\)
\(=\sqrt{3}-\sqrt{2}+\sqrt{2}+1=\sqrt{3}+1\)
5: \(A=\sqrt{13-4\sqrt{3}}+\sqrt{7-4\sqrt{3}}\)
\(=\sqrt{12-2\cdot2\sqrt{3}\cdot1+1}+\sqrt{4-2\cdot2\cdot\sqrt{3}+3}\)
\(=\sqrt{\left(2\sqrt{3}-1\right)^2}+\sqrt{\left(\sqrt{3}-1\right)^2}\)
\(=2\sqrt{3}-1+\sqrt{3}-1=3\sqrt{3}-2\)
6: \(A=\sqrt{5+2\sqrt{6}}-\sqrt{5-2\sqrt{6}}\)
\(=\sqrt{3+2\cdot\sqrt{3}\cdot\sqrt{2}+2}-\sqrt{3-2\cdot\sqrt{3}\cdot\sqrt{2}+2}\)
\(=\sqrt{\left(\sqrt{3}+\sqrt{2}\right)^2}-\sqrt{\left(\sqrt{3}-\sqrt{2}\right)^2}\)
\(=\sqrt{3}+\sqrt{2}-\sqrt{3}+\sqrt{2}=2\sqrt{2}\)
7: \(A=\sqrt{24+8\sqrt{5}}+\sqrt{9-4\sqrt{5}}\)
\(=\sqrt{20+2\cdot2\sqrt{5}\cdot2+4}+\sqrt{5-2\cdot\sqrt{5}\cdot2+4}\)
\(=\sqrt{\left(2\sqrt{5}+2\right)^2}+\sqrt{\left(\sqrt{5}-2\right)^2}\)
\(=2\sqrt{5}+2+\sqrt{5}-2=3\sqrt{5}\)
8: \(A=\sqrt{15-6\sqrt{6}}+\sqrt{33-12\sqrt{6}}\)
\(=\sqrt{9-2\cdot3\cdot\sqrt{6}+6}+\sqrt{24-2\cdot2\sqrt{6}\cdot3+9}\)
\(=\sqrt{\left(3-\sqrt{6}\right)^2}+\sqrt{\left(2\sqrt{6}-3\right)^2}\)
\(=3-\sqrt{6}+2\sqrt{6}-3=\sqrt{6}\)
9: \(A=\sqrt{6-4\sqrt{2}}+\sqrt{22-12\sqrt{2}}\)
\(=\sqrt{4-2\cdot2\cdot\sqrt{2}+2}+\sqrt{18-2\cdot3\sqrt{2}\cdot2+4}\)
\(=\sqrt{\left(2-\sqrt{2}\right)^2}+\sqrt{\left(3\sqrt{2}-2\right)^2}\)
\(=2-\sqrt{2}+3\sqrt{2}-2=2\sqrt{2}\)
10: \(A=\sqrt{17-12\sqrt{2}}+\sqrt{9+4\sqrt{2}}\)
\(=\sqrt{9-2\cdot3\cdot2\sqrt{2}+8}+\sqrt{8+2\cdot2\sqrt{2}\cdot1+1}\)
\(=\sqrt{\left(3-2\sqrt{2}\right)^2}+\sqrt{\left(2\sqrt{2}+1\right)^2}\)
\(=3-2\sqrt{2}+2\sqrt{2}+1=4\)
