\(\sqrt{5+2\sqrt{6}}-\sqrt{5-2\sqrt{6}}\)
\(=\sqrt{\left(\sqrt{3}\right)^2+2\cdot\sqrt{3}\cdot\sqrt{2}+\left(\sqrt{2}\right)^2}-\sqrt{\left(\sqrt{3}\right)^2-2\cdot\sqrt{3}\cdot\sqrt{2}+\left(\sqrt{2}\right)^2}\)
\(=\sqrt{\left(\sqrt{3}+\sqrt{2}\right)^2}-\sqrt{\left(\sqrt{3}-\sqrt{2}\right)^2}\)
\(=\left|\sqrt{3}+\sqrt{2}\right|-\left|\sqrt{3}-\sqrt{2}\right|\)
\(=\sqrt{3}+\sqrt{2}-\sqrt{3}+\sqrt{2}\)
\(=2\sqrt{2}\)
a: \(=\sqrt{\left(\sqrt{3}+\sqrt{2}\right)^2}-\sqrt{\left(\sqrt{3}-\sqrt{2}\right)^2}\)
\(=\sqrt{3}+\sqrt{2}-\sqrt{3}+\sqrt{2}=2\sqrt{2}\)
b: \(=\sqrt{\left(2+\sqrt{3}\right)^2}-\sqrt{5}+2-4+\sqrt{5}\)
\(=2+\sqrt{3}-2=\sqrt{3}\)
\(\sqrt{7+4\sqrt{3}}-\sqrt{\left(2-\sqrt{5}\right)^2}-\sqrt{21-8\sqrt{5}}\)
\(=\sqrt{2^2+2\cdot2\cdot\sqrt{3}+\left(\sqrt{3}\right)^2}-\left|2-\sqrt{5}\right|-\sqrt{4^2-2\cdot4\cdot\sqrt{5}+\left(\sqrt{5}\right)^2}\)
\(=\sqrt{\left(2+\sqrt{3}\right)^2}+\left(2-\sqrt{5}\right)-\sqrt{\left(4-\sqrt{5}\right)^2}\)
\(=\left|2+\sqrt{3}\right|+2-\sqrt{5}-\left|4-\sqrt{5}\right|\)
\(=2+\sqrt{3}+2-\sqrt{5}-4+\sqrt{5}\)
\(=\sqrt{3}\)
