1)
\(\sqrt{6+2\sqrt{5}}\\ =\sqrt{5+2\sqrt{5}+1}\\ =\sqrt{\left(\sqrt{5}+1\right)^2}\\ =\left|\sqrt{5}+1\right|\)
\(=\sqrt{5}+1\) (\(\sqrt{5}+1>0\) )
3)
\(\sqrt{15-6\sqrt{6}}-\sqrt{10-4\sqrt{6}}\\ =\sqrt{9-6\sqrt{6}+6}-\sqrt{6-4\sqrt{6}+4}\\ =\sqrt{\left(3-\sqrt{6}\right)^2}-\sqrt{\left(\sqrt{6}-2\right)^2}\\ =\left|3-\sqrt{6}\right|-\left|\sqrt{6}-2\right|\)
\(=3-\sqrt{6}-\left(\sqrt{6}-2\right)\) (vì \(3-\sqrt{6}>0;\sqrt{6}-2>0\) )
\(=3-\sqrt{6}-\sqrt{6}+2\\ =5-2\sqrt{6}\)
5)
\(\sqrt{31-10\sqrt{6}}-\sqrt{\left(3-2\sqrt{6}\right)^2}\\ =\sqrt{25-10\sqrt{6}+6}-\left|3-2\sqrt{6}\right|\)
\(=\sqrt{\left(5-\sqrt{6}\right)^2}-\left(2\sqrt{6}-3\right)\) (vì \(3-2\sqrt{6}< 0\) )
\(=\left|5-\sqrt{6}\right|-2\sqrt{6}+3\)
\(=5-\sqrt{6}-2\sqrt{6}+3\) (vì \(5-\sqrt{6}>0\) )
\(=8-3\sqrt{6}\)
1) \(\sqrt{6+2\sqrt{5}}=\sqrt{5+2\sqrt{5}+1}=\sqrt{\left(\sqrt{5}\right)^2+2\sqrt{5}+1^2}=\sqrt{\left(\sqrt{5}+1\right)^2}=\sqrt{5}+1\)
3) \(\sqrt{15-6\sqrt{6}}-\sqrt{10-4\sqrt{6}}\)
\(=\sqrt{15-2\cdot3\sqrt{6}}-\sqrt{10-2\cdot2\sqrt{6}}\)
\(=\sqrt{9-2\cdot3\sqrt{6}+6}-\sqrt{4-2\cdot2\sqrt{6}+6}\)
\(=\sqrt{3^2-2\cdot3\sqrt{6}+\left(\sqrt{6}\right)^2}-\sqrt{2^2-2\cdot2\sqrt{6}+\left(\sqrt{6}\right)^2}\)
\(=\sqrt{\left(3-\sqrt{6}\right)^2}-\sqrt{\left(2-\sqrt{6}\right)^2}\)
\(=\left|3-\sqrt{6}\right|-\left|2-\sqrt{6}\right|\)
\(=\left(3-\sqrt{6}\right)-\left(\sqrt{6}-2\right)\)
\(=3-\sqrt{6}-\sqrt{6}+2\)
\(=5-2\sqrt{6}\)
5) \(\sqrt{31-10\sqrt{6}}-\sqrt{\left(3-2\sqrt{6}\right)^2}\)
\(=\sqrt{25-2\cdot5\sqrt{6}+6}-\left|3-2\sqrt{6}\right|\)
\(=\sqrt{5^2-2\cdot5\sqrt{6}+\left(\sqrt{6}\right)^2}-\left(2\sqrt{6}-3\right)\)
\(=\sqrt{\left(5-\sqrt{6}\right)^2}-2\sqrt{6}+3\)
\(=\left|5-\sqrt{6}\right|-2\sqrt{6}+3\)
\(=5-\sqrt{6}-2\sqrt{6}+3\)
\(=8-3\sqrt{6}\)
