3) \(A=\sqrt[]{9+4\sqrt[]{2}}-\sqrt[]{9-4\sqrt[]{2}}\)
\(A=\sqrt[]{9+2.2\sqrt[]{2}}-\sqrt[]{9-2.2\sqrt[]{2}}\)
\(A=\sqrt[]{8+2.2\sqrt[]{2}+1}-\sqrt[]{8-2.2\sqrt[]{2}+1}\)
\(A=\sqrt[]{\left(2\sqrt[]{2}\right)^2+2.2\sqrt[]{2}+1^2}-\sqrt[]{\left(2\sqrt[]{2}\right)^2-2.2\sqrt[]{2}+1^2}\)
\(A=\sqrt[]{\left(2\sqrt[]{2}+1\right)^2}-\sqrt[]{\left(2\sqrt[]{2}-1\right)^2}\)
\(A=\left|2\sqrt[]{2}+1\right|-\left|2\sqrt[]{2}-1\right|\)
\(A=2\sqrt[]{2}+1-2\sqrt[]{2}+1\left(2\sqrt[]{2}>1\Rightarrow8>1\right)\)
\(A=2\)
\(3,\sqrt{9+4\sqrt{2}}-\sqrt{9-4\sqrt{2}}\\ =\sqrt{\left(2\sqrt{2}\right)^2+2\cdot2\sqrt{2}\cdot1+1^2}-\sqrt{\left(2\sqrt{2}\right)-2\cdot2\sqrt{2}\cdot1+1^2}\\ =\sqrt{\left(2\sqrt{2}+1\right)^2}-\sqrt{\left(2\sqrt{2}-1\right)^2}\\ =2\sqrt{2}+1-2\sqrt{2}+1\\ =2\\ 4,\sqrt{5-2\sqrt{6}}+\sqrt{3+2\sqrt{2}}\\ =\sqrt{\left(\sqrt{3}\right)^2-2\cdot\sqrt{3}\cdot\sqrt{2}+\left(\sqrt{2}\right)^2}+\sqrt{\left(\sqrt{2}\right)^2+2\cdot\sqrt{2}\cdot1+1^2}\\ =\sqrt{\left(\sqrt{3}-\sqrt{2}\right)^2}+\sqrt{\left(\sqrt{2}+1\right)^2}\\ =\sqrt{3}-\sqrt{2}+\sqrt{2}+1\\ =\sqrt{3}-1\)
\(5,\sqrt{13-4\sqrt{3}}+\sqrt{7-4\sqrt{3}}\\ =\sqrt{\left(2\sqrt{3}\right)^2-2\cdot2\sqrt{3}\cdot1+1^2}+\sqrt{2^2-2\cdot2\cdot\sqrt{3}+\left(\sqrt{3}\right)^2}\\ =\sqrt{\left(2\sqrt{3}-1\right)^2}+\sqrt{\left(2-\sqrt{3}\right)^2}\\ =2\sqrt{3}-1+2-\sqrt{3}\\ =1-\sqrt{3}\\ 6,\sqrt{5+2\sqrt{6}}-\sqrt{5-2\sqrt{6}}\\ =\sqrt{\left(\sqrt{3}\right)^2+2\cdot\sqrt{3}\cdot\sqrt{2}+\left(\sqrt{2}\right)^2}-\sqrt{\left(\sqrt{3}\right)^2-2\cdot\sqrt{3}\cdot\sqrt{2}+\left(\sqrt{2}\right)^2}\\ =\sqrt{\left(\sqrt{3}+\sqrt{2}\right)^2}-\sqrt{\left(\sqrt{3}-\sqrt{2}\right)^2}\\ =\sqrt{3}+\sqrt{2}-\sqrt{3}+\sqrt{2}\\ =2\sqrt{2}\)
3) \(A=\sqrt{9+4\sqrt{2}}-\sqrt{9-4\sqrt{2}}\)
\(A=\sqrt{8+2.2\sqrt{2}.1+1}-\sqrt{8-2.2\sqrt{2}.1+1}\)
\(A=\sqrt{\left(2\sqrt{2}+1\right)^2}-\sqrt{\left(2\sqrt{2}-1\right)^2}\)
\(A=\left|2\sqrt{2}+1\right|-\left|2\sqrt{2}-1\right|\)
\(A=2\sqrt{2}+1-\left(2\sqrt{2}-1\right)\)
\(A=2\sqrt{2}+1-2\sqrt{2}+1\)
\(A=2\)
4) \(A=\sqrt{5-2\sqrt{6}}+\sqrt{3+2\sqrt{2}}\)
\(A=\sqrt{3-2.\sqrt{3}.\sqrt{2}+2}+\sqrt{2+2.\sqrt{2}.1+1}\)
\(A=\sqrt{\left(\sqrt{3}-\sqrt{2}\right)^2}+\sqrt{\left(\sqrt{2}+1\right)^2}\)
\(A=\left|\sqrt{3}-\sqrt{2}\right|+\left|\sqrt{2}+1\right|\)
\(A=\sqrt{3}-\sqrt{2}+\sqrt{2}+1\)
\(A=\sqrt{3}+1\)
5) \(A=\sqrt{13-4\sqrt{3}}+\sqrt{7-4\sqrt{3}}\)
\(A=\sqrt{\left(2\sqrt{3}\right)^2-2.2\sqrt{3}.1+1}+\sqrt{2^2-2.2.\sqrt{3}+3}\)
\(A=\sqrt{\left(2\sqrt{3}-1\right)^2}+\sqrt{\left(2-\sqrt{3}\right)^2}\)
\(A=\left|2\sqrt{3}-1\right|+\left|2-\sqrt{3}\right|\)
\(A=2\sqrt{3}-1+2-\sqrt{3}\)
\(A=\sqrt{3}+1\)
6) \(A=\sqrt{5+2\sqrt{6}}-\sqrt{5-2\sqrt{6}}\)
\(A=\sqrt{2+2.\sqrt{2}.\sqrt{3}+3}-\sqrt{2-2\sqrt{3}.\sqrt{2}+3}\)
\(A=\sqrt{\left(\sqrt{2}+\sqrt{3}\right)^2}-\sqrt{\left(\sqrt{2}-\sqrt{3}\right)^2}\)
\(A=\left|\sqrt{2}+\sqrt{3}\right|-\left|\sqrt{2}-\sqrt{3}\right|\)
\(A=\sqrt{2}+\sqrt{3}-\left(\sqrt{3}-\sqrt{2}\right)\)
\(A=\sqrt{2}+\sqrt{3}-\sqrt{3}+\sqrt{2}\)
\(A=2\sqrt{2}\)