Giúp vs mn hứa cho 5* + CTLHN Thực hiện phép tính: $\sqrt{3+\sqrt{5}}$ + $\sqrt{7-3\sqrt{5}}$ - $\sqrt{2}$

Đáp án:

`sqrt{3+sqrt{5}}+sqrt{7-3sqrt{5}}-sqrt{2}=sqrt{2}`

Giải thích các bước giải:

`sqrt{3+sqrt{5}}+sqrt{7-3sqrt{5}}-sqrt{2}`
`=sqrt{(2.(3+sqrt{5}))/2}+sqrt{(2.(7-3sqrt{5}))/2}-sqrt{2}`
`=sqrt{(6+2sqrt{5})/2}+sqrt{(14-6sqrt{5})/2}-sqrt{2}`
`=sqrt{6+2sqrt{5}}/sqrt{2}+sqrt{14-6sqrt{5}}/sqrt{2}-sqrt{2}`
`=sqrt{(sqrt{5})^2+2.sqrt{5}.1+1^2}/sqrt{2}+sqrt{3^2-2.3.sqrt{5}+(sqrt{5})^2}/sqrt{2}-sqrt{2}`
`=sqrt{(sqrt{5}+1)^2}/sqrt{2}+sqrt{(3-sqrt{5})^2}/sqrt{2}-sqrt{2}`
`=|sqrt{5}+1|/sqrt{2}+|3-sqrt{5}|/sqrt{2}-sqrt{2}`
`=(sqrt{5}+1)/sqrt{2}+(3-sqrt{5})/sqrt{2}-sqrt{2}` ( vì `sqrt{9}(=3)>sqrt{5}`)
`=(sqrt{5}+1+3-sqrt{5})/sqrt{2}-sqrt{2}`
`=(4)/sqrt{2}-sqrt{2}`
`=(sqrt{2}.sqrt{2}.2)/sqrt{2}-sqrt{2}`
`=2sqrt{2}-sqrt{2}`
`=sqrt{2}`