1) \(\sqrt{11-6\sqrt{2}}-\sqrt{6-4\sqrt{2}}\)
\(=\sqrt{9-6\sqrt{2}+2}-\sqrt{4-4\sqrt{2}+2}\)
\(=\sqrt{3^2-2\cdot3\cdot\sqrt{2}+\left(\sqrt{2}\right)^2}-\sqrt{2^2-2\cdot2\cdot\sqrt{2}\cdot\left(\sqrt{2}\right)^2}\)
\(=\sqrt{\left(3-\sqrt{2}\right)^2}-\sqrt{\left(2-\sqrt{2}\right)^2}\)
\(=\left|3-\sqrt{2}\right|-\left|2-\sqrt{2}\right|\)
\(=3-\sqrt{2}-2+\sqrt{2}=1\)
2) \(\sqrt{\left(1-\sqrt{2}\right)^2}-\sqrt{\left(2-\sqrt{2}\right)^2}\)
\(=\left|1-\sqrt{2}\right|-\left|2-\sqrt{2}\right|\)
\(=\sqrt{2}-1+2-\sqrt{2}\)
\(=1\)
3) \(\sqrt{\left(2\sqrt{2}-1\right)^2}-\sqrt{17+12\sqrt{2}}\)
\(=\sqrt{\left(2\sqrt{2}-1\right)^2}-\sqrt{9+12\sqrt{2}+8}\)
\(=\sqrt{\left(2\sqrt{2}-1\right)^2}-\sqrt{3^2+2\cdot3\cdot2\sqrt{2}+\left(2\sqrt{2}\right)^2}\)
\(=\sqrt{\left(2\sqrt{2}-1\right)^2}-\sqrt{\left(3+2\sqrt{2}\right)^2}\)
\(=\left|2\sqrt{2}-1\right|-\left|3+2\sqrt{2}\right|\)
\(=2\sqrt{2}-1-3-2\sqrt{2}=-4\)
4) \(\sqrt{\left(2-\sqrt{5}\right)^2}+\sqrt{14-6\sqrt{5}}\)
\(=\sqrt{\left(2-\sqrt{5}\right)^2}+\sqrt{9-6\sqrt{5}+5}\)
\(=\sqrt{\left(2-\sqrt{5}\right)^2}+\sqrt{3^2-2\cdot3\cdot\sqrt{5}+\left(\sqrt{5}\right)^2}\)
\(=\sqrt{\left(2-\sqrt{5}\right)^2}+\sqrt{\left(3-\sqrt{5}\right)^2}\)
\(=\left|2-\sqrt{5}\right|+\left|3-\sqrt{5}\right|\)
\(=\sqrt{5}-2+3-\sqrt{5}=1\)
5) \(\sqrt{\left(4-3\sqrt{2}\right)^2}-\sqrt{19+6\sqrt{2}}\)
\(=\sqrt{\left(4-3\sqrt{2}\right)^2}-\sqrt{18+6\sqrt{2}+1}\)
\(=\sqrt{\left(4-3\sqrt{2}\right)^2}-\sqrt{\left(3\sqrt{2}\right)^2+2\cdot3\sqrt{2}\cdot1+1^2}\)
\(=\sqrt{\left(4-3\sqrt{2}\right)^2}-\sqrt{\left(3\sqrt{2}+1\right)^2}\)
\(=\left|4-3\sqrt{2}\right|-\left|3\sqrt{2}+1\right|\)
\(=3\sqrt{2}-4-3\sqrt{2}-1=-5\)
6) \(\sqrt{3-2\sqrt{2}}+\sqrt{\left(2-\sqrt{2}\right)^2}\)
\(=\sqrt{2-2\sqrt{2}+1}+\sqrt{\left(2-\sqrt{2}\right)^2}\)
\(=\sqrt{\left(\sqrt{2}\right)^2-2\cdot\sqrt{2}\cdot1+1^2}+\sqrt{\left(2-\sqrt{2}\right)^2}\)
\(=\sqrt{\left(\sqrt{2}-1\right)^2}+\sqrt{\left(2-\sqrt{2}\right)^2}\)
\(=\sqrt{2}-1+2-\sqrt{2}=1\)
1: =3-căn 2-2+căn 2=1
2: \(=\sqrt{2}-1+2-\sqrt{2}=1\)
3: \(=2\sqrt{2}-1-3-2\sqrt{2}=-4\)
4: \(=\sqrt{5}-2+3-\sqrt{5}=1\)
5: \(=3\sqrt{2}-4-3\sqrt{2}-1=-5\)
6: \(=\sqrt{2}-1+2-\sqrt{2}=1\)
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