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\(\frac{2+\sqrt{3}}{\sqrt{2}+\sqrt{2+\sqrt{3}}}+\frac{2-\sqrt{3}}{\sqrt{2}-\sqrt{2-\sqrt{3}}}\)\(=\frac{4+2\sqrt{3}}{\sqrt{4}+\sqrt{4+2\sqrt{3}}}+\frac{4-2\sqrt{3}}{\sqrt{4}-\sqrt{4-2\sqrt{3}}}\)
\(=\frac{4+2\sqrt{3}}{2+\sqrt{\left(\sqrt{3}+1\right)^2}}+\frac{4-2\sqrt{3}}{2-\sqrt{\left(\sqrt{3}-1\right)^2}}\)\(=\frac{4+2\sqrt{3}}{2+\sqrt{3}+1}+\frac{4-2\sqrt{3}}{2-\sqrt{3}+1}\)
\(=\frac{\left(\sqrt{3}+1\right)^2}{3+\sqrt{3}}+\frac{\left(\sqrt{3}-1\right)^2}{3-\sqrt{3}}\)
\(=\frac{\left(\sqrt{3}+1\right)^2}{\sqrt{3}\left(\sqrt{3}+1\right)}+\frac{\left(\sqrt{3}-1\right)^2}{\sqrt{3}\left(\sqrt{3}-1\right)}=\frac{\sqrt{3}+1}{\sqrt{3}}+\frac{\sqrt{3}-1}{\sqrt{3}}\)
\(=\frac{2\sqrt{3}}{\sqrt{3}}=2\)
\(=\frac{2-1}{\sqrt{2}+1}+\frac{3-2}{\sqrt{3}+\sqrt{2}}+\frac{4-3}{\sqrt{4}+\sqrt{3}}+...+\frac{100-99}{\sqrt{100}+\sqrt{99}}.\)
\(=\frac{\left(\sqrt{2}+1\right)\left(\sqrt{2}-1\right)}{\sqrt{2}+1}+\frac{\left(\sqrt{3}+\sqrt{2}\right)\left(\sqrt{3}-\sqrt{2}\right)}{\sqrt{3}+\sqrt{2}}+\frac{\left(\sqrt{4}+\sqrt{3}\right)\left(\sqrt{4}-\sqrt{3}\right)}{\sqrt{4}+\sqrt{3}}+...\)
\(=\sqrt{2}-1+\sqrt{3}-\sqrt{2}+\sqrt{4}-\sqrt{3}+...+\sqrt{100}-\sqrt{99}\)
\(=\sqrt{100}-1=10-1=9.\)
\(A=\frac{2+\sqrt{3}}{\sqrt{2}+\sqrt{2+\sqrt{3}}}+\frac{2-\sqrt{3}}{\sqrt{2}-\sqrt{2-\sqrt{3}}}\)
\(\Rightarrow\)\(\frac{A}{\sqrt{2}}=\frac{2+\sqrt{3}}{2+\sqrt{4+2\sqrt{3}}}+\frac{2-\sqrt{3}}{2-\sqrt{4-2\sqrt{3}}}\)
\(=\frac{2+\sqrt{3}}{2+\left(\sqrt{3}+1\right)}+\frac{2-\sqrt{3}}{2-\left(\sqrt{3}-1\right)}\)
\(=\frac{2+\sqrt{3}}{3+\sqrt{3}}+\frac{2-\sqrt{3}}{3-\sqrt{3}}\)
\(=\frac{\left(2+\sqrt{3}\right)\left(\sqrt{3}-1\right)}{\sqrt{3}\left(\sqrt{3}+1\right)\left(\sqrt{3}-1\right)}+\frac{\left(2-\sqrt{3}\right)\left(\sqrt{3}+1\right)}{\sqrt{3}\left(\sqrt{3}-1\right)\left(\sqrt{3}+1\right)}\)
\(=\frac{\sqrt{3}+1}{\sqrt{3}\left(\sqrt{3}-1\right)\left(\sqrt{3}+1\right)}+\frac{\sqrt{3}-1}{\sqrt{3}\left(\sqrt{3}-1\right)\left(\sqrt{3}+1\right)}\)
\(=\frac{2\sqrt{3}}{2\sqrt{3}}=1\)