1: \(A=\sqrt{6-2\sqrt{5}}+\sqrt{9-4\sqrt{5}}\)
\(=\sqrt{5-2\cdot\sqrt{5}\cdot1+1}+\sqrt{5-2\cdot\sqrt{5}\cdot2+4}\)
\(=\sqrt{\left(\sqrt{5}-1\right)^2}+\sqrt{\left(\sqrt{5}-2\right)^2}\)
\(=\sqrt{5}-1+\sqrt{5}-2=2\sqrt{5}-3\)
2: \(A=\sqrt{4-2\sqrt{3}}+\sqrt{4+2\sqrt{3}}\)
\(=\sqrt{3-2\cdot\sqrt{3}\cdot1+1}+\sqrt{3+2\cdot\sqrt{3}\cdot1+1}\)
\(=\sqrt{\left(\sqrt{3}-1\right)^2}+\sqrt{\left(\sqrt{3}+1\right)^2}\)
\(=\sqrt{3}-1+\sqrt{3}+1=2\sqrt{3}\)
3: \(A=\sqrt{7-4\sqrt{3}}-\sqrt{4-2\sqrt{3}}\)
\(=\sqrt{4-2\cdot2\cdot\sqrt{3}+3}-\sqrt{3-2\cdot\sqrt{3}\cdot1+1}\)
\(=\sqrt{\left(2-\sqrt{3}\right)^2}-\sqrt{\left(\sqrt{3}-1\right)^2}\)
\(=2-\sqrt{3}-\sqrt{3}+1=3-2\sqrt{3}\)
4: \(A=\sqrt{4+2\sqrt{3}}-\sqrt{4-2\sqrt{3}}\)
\(=\sqrt{3+2\cdot\sqrt{3}\cdot1+1}-\sqrt{3-2\cdot\sqrt{3}\cdot1+1}\)
\(=\sqrt{\left(\sqrt{3}+1\right)^2}-\sqrt{\left(\sqrt{3}-1\right)^2}\)
\(=\sqrt{3}+1-\sqrt{3}+1=2\)
5: \(A=\sqrt{7-2\sqrt{6}}-\sqrt{7+2\sqrt{6}}\)
\(=\sqrt{6-2\cdot\sqrt{6}\cdot1+1}-\sqrt{6+2\cdot\sqrt{6}\cdot1+1}\)
\(=\sqrt{\left(\sqrt{6}-1\right)^2}-\sqrt{\left(\sqrt{6}+1\right)^2}\)
\(=\sqrt{6}-1-\sqrt{6}-1=-2\)
6: \(A=\sqrt{3+2\sqrt{2}}+\sqrt{3-2\sqrt{2}}\)
\(=\sqrt{2+2\cdot\sqrt{2}\cdot1+1}+\sqrt{2-2\cdot\sqrt{2}\cdot1+1}\)
\(=\sqrt{\left(\sqrt{2}+1\right)^2}+\sqrt{\left(\sqrt{2}-1\right)^2}\)
\(=\sqrt{2}+1+\sqrt{2}-1=2\sqrt{2}\)
7: \(A=\sqrt{11+6\sqrt{2}}-\sqrt{11-6\sqrt{2}}\)
\(=\sqrt{9+2\cdot3\cdot\sqrt{2}+2}-\sqrt{9-2\cdot3\cdot\sqrt{2}+2}\)
\(=\sqrt{\left(3+\sqrt{2}\right)^2}-\sqrt{\left(3-\sqrt{2}\right)^2}\)
\(=3+\sqrt{2}-3+\sqrt{2}=2\sqrt{2}\)
8: \(A=\sqrt{8-2\sqrt{7}}-\sqrt{8+2\sqrt{7}}\)
\(=\sqrt{7-2\cdot\sqrt[]{7}\cdot1+1}-\sqrt{7+2\cdot\sqrt{7}\cdot1+1}\)
\(=\sqrt{\left(\sqrt{7}-1\right)^2}-\sqrt{\left(\sqrt{7}+1\right)^2}\)
\(=\sqrt{7}-1-\sqrt{7}-1=-2\)
9: \(A=\sqrt{18+8\sqrt{2}}+\sqrt{18-8\sqrt{2}}\)
\(=\sqrt{16+2\cdot4\cdot\sqrt{2}+2}+\sqrt{16-2\cdot4\cdot\sqrt{2}+2}\)
\(=\sqrt{\left(4+\sqrt{2}\right)^2}+\sqrt{\left(4-\sqrt{2}\right)^2}\)
\(=4+\sqrt{2}+4-\sqrt{2}=8\)
10: \(A=\sqrt{28+10\sqrt{3}}+\sqrt{19-8\sqrt{3}}\)
\(=\sqrt{25+2\cdot5\cdot\sqrt{3}+3}+\sqrt{16-2\cdot4\cdot\sqrt{3}+3}\)
\(=\sqrt{\left(5+\sqrt{3}\right)^2}+\sqrt{\left(4-\sqrt{3}\right)^2}\)
\(=5+\sqrt{3}+4-\sqrt{3}=9\)