Đáp án:
`a)` `{3\sqrt{3}}/{\sqrt{2}+\sqrt{3}+\sqrt{5}}={3.(2+\sqrt{6}-\sqrt{10})}/4`
`b)` `{\sqrt{2}+\sqrt{3}+\sqrt{4}}/{\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+4}=\sqrt{2}-1`
`c)` `{\sqrt{\sqrt{5}+\sqrt{2}}}/{\sqrt{3\sqrt{5}-3\sqrt{2}}}={\sqrt{5}+\sqrt{2}}/3`
Giải thích các bước giải:
`a)` `{3\sqrt{3}}/{\sqrt{2}+\sqrt{3}+\sqrt{5}} `
`={3\sqrt{3}.(\sqrt{2}+\sqrt{3}-\sqrt{5})}/{(\sqrt{2}+\sqrt{3}+\sqrt{5}).(\sqrt{2}+\sqrt{3}-\sqrt{5})}`
`={3\sqrt{3}.(\sqrt{2}+\sqrt{3}-\sqrt{5})}/{(\sqrt{2}+\sqrt{3})^2-(\sqrt{5})^2}`
`={3\sqrt{3}.(\sqrt{2}+\sqrt{3}-\sqrt{5})}/{(\sqrt{2}+\sqrt{3})^2-(\sqrt{5})^2}`
`={3\sqrt{3}.(\sqrt{2}+\sqrt{3}-\sqrt{5})}/{2+2\sqrt{2}.\sqrt{3}+3-5}`
`={3\sqrt{3}.(\sqrt{2}+\sqrt{3}-\sqrt{5})}/{2\sqrt{2}.\sqrt{3}}`
`={3.(\sqrt{2}+\sqrt{3}-\sqrt{5})}/{2\sqrt{2}}`
`={3.(\sqrt{2}+\sqrt{3}-\sqrt{5}).\sqrt{2}}/{2\sqrt{2}.\sqrt{2}}`
`={3.(2+\sqrt{6}-\sqrt{10})}/4`
Vậy: `{3\sqrt{3}}/{\sqrt{2}+\sqrt{3}+\sqrt{5}}={3.(2+\sqrt{6}-\sqrt{10})}/4`
$\\$
`b)` `{\sqrt{2}+\sqrt{3}+\sqrt{4}}/{\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+4}`
`={\sqrt{2}+\sqrt{3}+\sqrt{4}}/{\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+\sqrt{4}+\sqrt{4}}`
`={\sqrt{2}+\sqrt{3}+\sqrt{4}}/{(\sqrt{2}+\sqrt{3}+\sqrt{4})+(\sqrt{4}+\sqrt{6}+\sqrt{8})}`
`={\sqrt{2}+\sqrt{3}+\sqrt{4}}/{(\sqrt{2}+\sqrt{3}+\sqrt{4})+\sqrt{2}.(\sqrt{2}+\sqrt{3}+\sqrt{4})}`
`={\sqrt{2}+\sqrt{3}+\sqrt{4}}/{(\sqrt{2}+\sqrt{3}+\sqrt{4}).(1+\sqrt{2})}`
`=1/{1+\sqrt{2}}`
`={\sqrt{2}-1}/{(\sqrt{2}-1).(\sqrt{2}+1)}`
`={\sqrt{2}-1}/{(\sqrt{2})^2-1^2}=\sqrt{2}-1`
Vậy: `{\sqrt{2}+\sqrt{3}+\sqrt{4}}/{\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+4}=\sqrt{2}-1`
$\\$
`c)` `{\sqrt{\sqrt{5}+\sqrt{2}}}/{\sqrt{3\sqrt{5}-3\sqrt{2}}}`
`={\sqrt{\sqrt{5}+\sqrt{2}}.\sqrt{\sqrt{5}+\sqrt{2}}}/{\sqrt{3\sqrt{5}-3\sqrt{2}}.\sqrt{\sqrt{5}+\sqrt{2}}}`
`={\sqrt{5}+\sqrt{2}}/{\sqrt{3.(\sqrt{5}-\sqrt{2}).(\sqrt{5}+\sqrt{2})}}`
`={\sqrt{5}+\sqrt{2}}/{\sqrt{3.(5-2)}}`
`={\sqrt{5}+\sqrt{2}}/3`
Vậy: `{\sqrt{\sqrt{5}+\sqrt{2}}}/{\sqrt{3\sqrt{5}-3\sqrt{2}}}={\sqrt{5}+\sqrt{2}}/3`
