`\sqrt{4+\sqrt{8}}.\sqrt{2+\sqrt{2+\sqrt{2}}}.\sqrt{2-\sqrt{2+\sqrt{2}}}`
`=\sqrt{(4+\sqrt{8})(2+\sqrt{2+\sqrt{2}})(2-\sqrt{2+\sqrt{2}})}`
`=\sqrt{(4+2\sqrt{2})(2+\sqrt{2+\sqrt{2}})(2-\sqrt{2+\sqrt{2}})}`
`=\sqrt{(4+2\sqrt{2})(4-(2+\sqrt{2}))}`
`=\sqrt{(4+2\sqrt{2})(4-2-\sqrt{2})}`
`=\sqrt{(4+2\sqrt{2})(2-\sqrt{2})}`
`=\sqrt{8-4\sqrt{2}+4\sqrt{2}-4}`
`=\sqrt{4}`
`=2`
