Lời giải.
`6.`
`\sqrt{3-\sqrt{5}}(\sqrt{10}-\sqrt{2})(3+\sqrt{5})`
`=\sqrt{3-\sqrt{5}}(\sqrt{2.5}-\sqrt{2.1})(3+\sqrt{5})`
`=\sqrt{3-\sqrt{5}}.\sqrt{2}.(\sqrt{5}-1)(3+\sqrt{5})`
`=\sqrt{2(3-\sqrt{5})}.(\sqrt{5}-1)(3+\sqrt{5})`
`=\sqrt{6-2\sqrt{5}}.(\sqrt{5}-1)(3+\sqrt{5})`
`=\sqrt{(\sqrt{5})^2-2.\sqrt{5}.1+1^2}.(\sqrt{5}-1)(3+\sqrt{5})`
`=\sqrt{(\sqrt{5}-1)^2}.(\sqrt{5}-1)(3+\sqrt{5})`
`=(\sqrt{5}-1).(\sqrt{5}-1)(3+\sqrt{5})`
`=(\sqrt{5}-1)^2(3+\sqrt{5}).`
`7.`
`\sqrt{\sqrt{5}-\sqrt{3-\sqrt{29-12\sqrt{5}}}}`
`=\sqrt{\sqrt{5}-\sqrt{3-\sqrt{29-2.3.\sqrt{4}.\sqrt{5}}}}`
`=\sqrt{\sqrt{5}-\sqrt{3-\sqrt{29-2.3.\sqrt{20}}}}`
`=\sqrt{\sqrt{5}-\sqrt{3-\sqrt{(\sqrt{20})^2-2.3.\sqrt{20}+3^2}}}`
`=\sqrt{\sqrt{5}-\sqrt{3-\sqrt{(\sqrt{20}-3)^2}}}`
`=\sqrt{\sqrt{5}-\sqrt{3-(\sqrt{20}-3)}}`
`=\sqrt{\sqrt{5}-\sqrt{3-\sqrt{4.5}+3}}`
`=\sqrt{\sqrt{5}-\sqrt{6-2\sqrt{5}}}`
`=\sqrt{\sqrt{5}-\sqrt{(\sqrt{5})^2-2.\sqrt{5}.1+1^2}}`
`=\sqrt{\sqrt{5}-\sqrt{(\sqrt{5}-1)^2}}`
`=\sqrt{\sqrt{5}-(\sqrt{5}-1)}`
`=\sqrt{\sqrt{5}-\sqrt{5}+1}`
`=\sqrt{1}=1.`
`8.`
`\sqrt{\sqrt{3}-\sqrt{1-\sqrt{21-12\sqrt{3}}}}`
`=\sqrt{\sqrt{3}-\sqrt{1-\sqrt{21-2.3.2.\sqrt{3}}}}`
`=\sqrt{\sqrt{3}-\sqrt{1-\sqrt{(2\sqrt{3})^2-2.3.2.\sqrt{3}+3^2}}}`
`=\sqrt{\sqrt{3}-\sqrt{1-\sqrt{(2\sqrt{3}-3)^2}}}`
`=\sqrt{\sqrt{3}-\sqrt{1-(2\sqrt{3}-3)}}`
`=\sqrt{\sqrt{3}-\sqrt{1-2\sqrt{3}+3}}`
`=\sqrt{\sqrt{3}-\sqrt{(\sqrt{3})^2-2\sqrt{3}+1^2}}`
`=\sqrt{\sqrt{3}-\sqrt{(\sqrt{3}-1)^2}}`
`=\sqrt{\sqrt{3}-(\sqrt{3}-1)}`
`=\sqrt{\sqrt{3}-\sqrt{3}+1}`
`=\sqrt{1}=1.`
`9.`
Ta có: `\sqrt{45+4\sqrt{41}}`
`=\sqrt{41+4\sqrt{41}+4}`
`=\sqrt{(\sqrt{41})^2+2.\sqrt{41}.2+2^2}`
`=\sqrt{(\sqrt{41}+1)^2}`
`=\sqrt{41}+1.`
Tương tự: `\sqrt{45-4\sqrt{41}}=\sqrt{(\sqrt{41}-1)^2}=\sqrt{41}-1.`
`=>\sqrt{45+4\sqrt{41}}+\sqrt{45-4\sqrt{41}}=\sqrt{41}+1+\sqrt{41}-1=2\sqrt{41}.`
`=>{8\sqrt{41}}/{\sqrt{45+4\sqrt{41}}+\sqrt{45-4\sqrt{41}}}={8\sqrt{41}}/{2\sqrt{41}}=4.`
`10.`
`\sqrt{13-30.\sqrt{2+\sqrt{9+4\sqrt{2}}}}`
`=\sqrt{13-30.\sqrt{2+\sqrt{8+1+2.2\sqrt{2}.1}}}`
`=\sqrt{13-30.\sqrt{2+\sqrt{(2\sqrt{2})^2+2.2\sqrt{2}.1+1^2}}}`
`=\sqrt{13-30.\sqrt{2+\sqrt{(2\sqrt{2}+1)^2}}}`
`=\sqrt{13-30.\sqrt{2+2\sqrt{2}+1}}`
`=\sqrt{13-30.\sqrt{(\sqrt{2})^2+2.\sqrt{2}.1+1^2}}`
`=\sqrt{13-30.\sqrt{(\sqrt{2}+1)^2}}`
`=\sqrt{13-30.(\sqrt{2}+1)}`
`=\sqrt{13-30.\sqrt{2}+30}`
`=\sqrt{43-30.\sqrt{2}}`
`=\sqrt{25-2.5.3.\sqrt{2}+18}`
`=\sqrt{5^2-2.5.3.\sqrt{2}+(3.\sqrt{2})^2}`
`=\sqrt{(5-3\sqrt{2})^2}`
`=5-3\sqrt{2}.`
