Đáp án+Giải thích các bước giải:
`1)sqrt{4+sqrt{15}}+sqrt{4-sqrt{15}}-2sqrt{3-sqrt5}`
`=sqrt{(8+2sqrt{15})/2}+sqrt{(8-2sqrt{15})/2}-2sqrt{(6-2sqrt5)/2}`
`=sqrt{(5+2sqrt{5.3}+3)/2}+sqrt{(5-2sqrt{5.3}+3)/2}-2sqrt{(5-2sqrt5+1)/2}`
`=sqrt{(sqrt5+sqrt3)^2/2}+sqrt{(sqrt5-sqrt3)^2/2}-2sqrt{(sqrt5-1)^2/2}`
`=(sqrt5+sqrt3)/sqrt2+(sqrt5-sqrt3)/sqrt2-2((sqrt5-1)/sqrt2)`
`=(sqrt5+sqrt3+sqrt5-sqrt3)/sqrt2-sqrt2(sqrt5-1)`
`=sqrt{10}-sqrt{10}+sqrt2=sqrt2`
`2)sqrt{5+sqrt{21}}-sqrt{5-sqrt{21}}`
`=sqrt{(10+2sqrt{21})/2}-sqrt{(10-2sqrt{21})/2}`
`=sqrt{(7+2sqrt{7.3}+3)/2}-sqrt{(7-2sqrt{7.3}+3)/2}`
`=sqrt{(sqrt7+sqrt3)^2/2}-sqrt{(sqrt7-sqrt3)^2/2}`
`=(sqrt7+sqrt3)/sqrt2-(sqrt7-sqrt3)/sqrt2`
`=(sqrt7+sqrt3-sqrt7+sqrt3)/sqrt2`
`=(2sqrt3)/sqrt2=sqrt6`
`3)sqrt{9-3sqrt5}-sqrt{9+3sqrt5}`
`=sqrt{3(3-sqrt5)}-sqrt{3(3+sqrt5)}`
`=sqrt{(3(6-2sqrt5))/2}-sqrt{(3(6+2sqrt5))/2}`
`=sqrt{(3(sqrt5-1)^2)/2}-sqrt{(3(sqrt5+1)^2)/2}`
`=(sqrt3(sqrt5-1))/sqrt2-(sqrt3(sqrt5+1))/sqrt2`
`=(sqrt{15}-sqrt3-sqrt{15}-sqrt3)/sqrt2`
`=(-2sqrt3)/sqrt2=-sqrt6`
`4)sqrt{8-3sqrt7}+sqrt{4-sqrt7}`
`=sqrt{(16-6sqrt7)/2}+sqrt{(8-2sqrt7)/2}`
`=sqrt{(9-2.3.sqrt7+7)/2}+sqrt{(7-2sqrt7+1)/2}`
`=sqrt{(3-sqrt7)^2/2}+sqrt{(sqrt7-1)^2/2}`
`=(3-sqrt7)/sqrt2+(sqrt7-1)/sqrt2`
`=(3-sqrt7+sqrt7-1)/sqrt2`
`=2/sqrt2=sqrt2`
`5)sqrt{6+sqrt{11}}-sqrt{6-sqrt{11}}+3sqrt2`
`=sqrt{(12+2sqrt{11})/2}-sqrt{(12-2sqrt{11})/2}+3sqrt2`
`=sqrt{(sqrt{11}+1)^2/2}-sqrt{(sqrt{11}-1)^2/2}+3sqrt2`
`=(sqrt{11}+1)/sqrt2-(sqrt{11}-1)/sqrt2+3sqrt2`
`=(sqrt{11}+1-sqrt{11}+2)/sqrt2+3sqrt2`
`=2/sqrt2+3sqrt2`
`=sqrt2+3sqrt2=4sqrt2`
