Đáp án:
`a)` `1`
`b)` `8`
Giải thích các bước giải:
`a)` `\sqrt{2+\sqrt{3}}.\sqrt{2+\sqrt{2+\sqrt{3}}}.\sqrt{2-\sqrt{2+\sqrt{3}}}`
`=\sqrt{2+\sqrt{3}}.\sqrt{(2+\sqrt{2+\sqrt{3}}).(2-\sqrt{2+\sqrt{3}})}`
`=\sqrt{2+\sqrt{3}}.\sqrt{2^2-(2+\sqrt{3})}`
`=\sqrt{2+\sqrt{3}}.\sqrt{2-\sqrt{3}}`
`=\sqrt{(2+\sqrt{3}).(2-\sqrt{3})}`
`=\sqrt{2^2-3}=\sqrt{1}=1`
Vậy: `\sqrt{2+\sqrt{3}}.\sqrt{2+\sqrt{2+\sqrt{3}}}.\sqrt{2-\sqrt{2+\sqrt{3}}}=1`
$\\$
`b)` `\sqrt{3-\sqrt{5}}.(\sqrt{10}-\sqrt{2}).(3+\sqrt{5})`
`=\sqrt{3-\sqrt{5}}.\sqrt{3+\sqrt{5}}.\sqrt{3+\sqrt{5}}.\sqrt{2}(\sqrt{5}-1)`
`=\sqrt{(3-\sqrt{5}).(3+\sqrt{5})}.\sqrt{2.(3+\sqrt{5})}.(\sqrt{5}-1)`
`=\sqrt{3^2-5}.\sqrt{5+2.\sqrt{5}.1+1}.(\sqrt{5}-1)`
`=\sqrt{4}.\sqrt{(\sqrt{5}+1)^2}.(\sqrt{5}-1)`
`=2.|\sqrt{5}+1|.(\sqrt{5}-1)`
`=2.(\sqrt{5}+1).(\sqrt{5}-1)`
`=2.(5-1^2)=2.4=8`
Vậy: `\sqrt{3-\sqrt{5}}.(\sqrt{10}-\sqrt{2}).(3+\sqrt{5})=8`
