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\(A=\sqrt{\frac{5+\sqrt{21}}{5-\sqrt{21}}}+\sqrt{\frac{5-\sqrt{21}}{5+\sqrt{21}}}\)
\(=\sqrt{\frac{\left(5+\sqrt{21}\right)^2}{\left(5-\sqrt{21}\right)\left(5+\sqrt{21}\right)}}+\sqrt{\frac{\left(5-\sqrt{21}\right)^2}{\left(5-\sqrt{21}\right)\left(5+\sqrt{21}\right)}}\)
\(=\sqrt{\frac{\left(5+\sqrt{21}\right)^2}{4}}+\sqrt{\frac{\left(5-\sqrt{21}\right)^2}{4}}\)
\(=\frac{5+\sqrt{21}}{2}+\frac{5-\sqrt{21}}{2}=5\)
\(B=\sqrt{7+\sqrt{33}}+\sqrt{7-\sqrt{33}}\)
\(\Rightarrow\)\(\sqrt{2}B=\sqrt{14+2\sqrt{33}}+\sqrt{14-2\sqrt{33}}\)
\(=\sqrt{\left(\sqrt{11}+\sqrt{3}\right)^2}+\sqrt{\left(\sqrt{11}-\sqrt{3}\right)^2}\)
\(=\sqrt{11}+\sqrt{3}+\sqrt{11}-\sqrt{3}=2\sqrt{11}\)
\(\Rightarrow\)\(B=\sqrt{22}\)
\(A=\sqrt{5+\sqrt{21}}+\sqrt{5-\sqrt{21}}\)
\(A^2=\left(\sqrt{5+\sqrt{21}}^2+2\sqrt{\left(5+\sqrt{21}\right)\left(5-\sqrt{21}\right)}+\sqrt{5-\sqrt{21}}^2\right)\)
\(A^2=5+\sqrt{21}+\sqrt{4\left(5+\sqrt{21}\right)\left(5-\sqrt{21}\right)}+5-\sqrt{21}\)
\(A^2=10+\sqrt{4.\left(25-21\right)}\)
\(A^2=10+\sqrt{4.4}=10+4=14\)
\(A=\sqrt{14}\)
Tương tự
1. \(\sqrt{7+2\sqrt{10}}-\sqrt{7-2\sqrt{10}}=\sqrt{\left(\sqrt{5}+\sqrt{2}\right)^2}-\sqrt{\left(\sqrt{5}-\sqrt{2}\right)^2}\\ =\sqrt{5}+\sqrt{2}-\sqrt{5}+\sqrt{2}=2\sqrt{2}\)
2. \(\sqrt{12-6\sqrt{3}}+\sqrt{21-12\sqrt{3}}=\sqrt{\left(3-\sqrt{3}\right)^2}+\sqrt{\left(2\sqrt{3}-3\right)^2}\\ =3-\sqrt{3}+2\sqrt{3}-3=\sqrt{3}\)
3. \(\sqrt{33-12\sqrt{6}}+\sqrt{15-6\sqrt{6}}=\sqrt{\left(2\sqrt{6}-3\right)^2}+\sqrt{\left(3+\sqrt{6}\right)^2}\\ =2\sqrt{6}-3+3+\sqrt{6}=3\sqrt{6}\)
1.\(\sqrt{7+2\sqrt{10}}-\sqrt{7-2\sqrt{10}}=\sqrt{\left(\sqrt{2}+\sqrt{5}\right)^2}-\sqrt{\left(\sqrt{5}-\sqrt{2}\right)^2}\)
\(=\sqrt{5}+\sqrt{2}-\left(\sqrt{5}-\sqrt{2}\right)=2\sqrt{2}\)
2. \(\sqrt{12-6\sqrt{3}+\sqrt{21-12\sqrt{3}}}=\sqrt{12-6\sqrt{3}+\sqrt{\left(3-2\sqrt{3}\right)^2}}\)
\(=\sqrt{12-6\sqrt{3}+2\sqrt{3}-3}=\sqrt{9-4\sqrt{3}}\)
3. \(\sqrt{33-12\sqrt{6}}+\sqrt{15-6\sqrt{6}}=\sqrt{\left(2\sqrt{6}-3\right)^2}+\sqrt{\left(\sqrt{6}-3\right)^2}\)
\(=2\sqrt{6}-3+3-\sqrt{6}=\sqrt{6}\)
a) \(\left(\sqrt{27}-\sqrt{12}-\sqrt{108}-\sqrt{192}\right):\sqrt{3}=\left(3\sqrt{3}-2\sqrt{3}-6\sqrt{3}-8\sqrt{3}\right):\sqrt{3}=\left(-13\sqrt{3}\right):\sqrt{3}=-13\sqrt{3}.\frac{1}{\sqrt{3}}=-13\)
c) \(\sqrt{15-6\sqrt{6}}+\sqrt{33-12\sqrt{6}}=\sqrt{\left(3-\sqrt{6}\right)^2}+\sqrt{\left(2\sqrt{6}-3\right)^2}=\left|3-\sqrt{6}\right|+\left|2\sqrt{6}-3\right|=3-\sqrt{6}+2\sqrt{6}-3=\sqrt{6}\)
a, \(=\left(3\sqrt{3}-2\sqrt{3}-6\sqrt{3}-8\sqrt{3}\right):\sqrt{3}\)
\(=\frac{-13\sqrt{3}}{\sqrt{3}}=-13\)
b, \(=\frac{\sqrt{2}\left(1-\sqrt{3}\right)}{1-\sqrt{3}}.\frac{3+2\sqrt{7}}{1+\sqrt{3}}\)
\(=\frac{\sqrt{2}\left(3+2\sqrt{7}\right)}{1+\sqrt{3}}\)
c, \(=\sqrt{6-6\sqrt{6} +9}+\sqrt{24-2.2\sqrt{6}.3+9}\)
\(=\sqrt{\left(\sqrt{6}-3\right)^2}+\sqrt{\left(2\sqrt{6}-3\right)^2}\)
\(=3-\sqrt{6}+2\sqrt{6}-3=\sqrt{6}\)
\(A=\sqrt{7+\sqrt{33}}+\sqrt{7-\sqrt{33}}>0\)
\(\Rightarrow A^2=14+2\sqrt{7^2-33}\)
\(\Rightarrow A^2=14+2\sqrt{16}=22\)
\(\Rightarrow A=\sqrt{22}\)