1: \(A=\sqrt{3-\sqrt{5}}+\sqrt{3+\sqrt{5}}\)
\(=\dfrac{\left(\sqrt{6-2\sqrt{5}}+\sqrt{6+2\sqrt{5}}\right)}{\sqrt{2}}\)
\(=\dfrac{\left(\sqrt{\left(\sqrt{5}-1\right)^2}+\sqrt{\left(\sqrt{5}+1\right)^2}\right)}{\sqrt{2}}\)
\(=\dfrac{\sqrt{5}-1+\sqrt{5}+1}{\sqrt{2}}=\dfrac{2\sqrt{5}}{\sqrt{2}}=\sqrt{10}\)
2: \(A=\sqrt{2+\sqrt{3}}-\sqrt{2-\sqrt{3}}\)
\(=\dfrac{\left(\sqrt{4+2\sqrt{3}}-\sqrt{4-2\sqrt{3}}\right)}{\sqrt{2}}\)
\(=\dfrac{\left(\sqrt{\left(\sqrt{3}+1\right)^2}-\sqrt{\left(\sqrt{3}-1\right)^2}\right)}{\sqrt{2}}\)
\(=\dfrac{\sqrt{3}+1-\sqrt{3}+1}{\sqrt{2}}=\dfrac{2}{\sqrt{2}}=\sqrt{2}\)
3: \(A=\sqrt{4+\sqrt{7}}-\sqrt{4-\sqrt{7}}\)
\(=\dfrac{\left(\sqrt{8+2\sqrt{7}}-\sqrt{8-2\sqrt{7}}\right)}{\sqrt{2}}\)
\(=\dfrac{\left(\sqrt{\left(\sqrt{7}+1\right)^2}-\sqrt{\left(\sqrt{7}-1\right)^2}\right)}{\sqrt{2}}\)
\(=\dfrac{\sqrt{7}+1-\sqrt{7}+1}{\sqrt{2}}=\dfrac{2}{\sqrt{2}}=\sqrt{2}\)
4: \(A=\sqrt{6-\sqrt{11}}+\sqrt{6+\sqrt{11}}\)
\(=\dfrac{\left(\sqrt{12-2\sqrt{11}}+\sqrt{12+2\sqrt{11}}\right)}{\sqrt{2}}\)
\(=\dfrac{\left(\sqrt{\left(\sqrt{11}-1\right)^2}+\sqrt{\left(\sqrt{11}+1\right)^2}\right)}{\sqrt{2}}\)
\(=\dfrac{\sqrt{11}-1+\sqrt{11}+1}{\sqrt{2}}=\dfrac{2\sqrt{11}}{\sqrt{2}}=\sqrt{22}\)
5: \(A=\sqrt{4-\sqrt{15}}-\sqrt{4+\sqrt{15}}\)
\(=\dfrac{\left(\sqrt{8-2\sqrt{15}}-\sqrt{8+2\sqrt{15}}\right)}{\sqrt{2}}\)
\(=\dfrac{\left(\sqrt{\left(\sqrt{5}-\sqrt{3}\right)^2}-\sqrt{\left(\sqrt{5}+\sqrt{3}\right)^2}\right)}{\sqrt{2}}\)
\(=\dfrac{\sqrt{5}-\sqrt{3}-\sqrt{5}-\sqrt{3}}{\sqrt[]{2}}=-\dfrac{2\sqrt{3}}{\sqrt{2}}=-\sqrt{6}\)
6: \(A=\sqrt{5-\sqrt{21}}-\sqrt{5+\sqrt{21}}\)
\(=\dfrac{\sqrt{10-2\sqrt{21}}-\sqrt[]{10+2\sqrt{21}}}{\sqrt{2}}\)
\(=\dfrac{\sqrt{\left(\sqrt{7}-\sqrt{3}\right)^2}-\sqrt{\left(\sqrt{7}+\sqrt{3}\right)^2}}{\sqrt{2}}\)
\(=\dfrac{\sqrt{7}-\sqrt{3}-\sqrt{7}-\sqrt{3}}{\sqrt{2}}=-\dfrac{2\sqrt{3}}{\sqrt{2}}=-\sqrt{6}\)
