Ta có: `x=4y+\sqrt{2xy}` (x, y > 0)
`<=> x-\sqrt{2xy}-4y=0`
`<=> x+\sqrt{2xy}-2\sqrt{2xy}-4y=0`
`<=> \sqrt{x}*(\sqrt{x}+\sqrt{2y})-2\sqrt{2y}*(\sqrt{x}+\sqrt{2y})=0`
`<=> (\sqrt{x}+\sqrt{2y})(\sqrt{x}-2\sqrt{2y})=0`
`<=> \sqrt{x}=2\sqrt{2y}`(do `\sqrt{x}+\sqrt{2y} > 0` với mọi x, y > 0)
`<=> x=8y`
Khi đó:
`P=(\sqrt[3]{xy}(3\sqrt[3]{x}-2\sqrt[3]{y}))/(\sqrt{2xy})`
`=(\sqrt[3]{8y*y}(3\sqrt[3]{8y}-2\sqrt[3]{y}))/(\sqrt{2*8y*y})`
`=(2\sqrt[3]{y^2}(6\sqrt[3]{y}-2\sqrt[3]{y}))/(\sqrt{16y^2})`
`=(2\sqrt[3]{y^2}*4\sqrt[3]{y})/(4y)`
`=(8y)/(4y)=2`
#`Toru`
Mình sửa lại do đánh lỗi một số chỗ nhé:
Ta có: `x=4y+\sqrt{2xy}` (x, y > 0)
`<=> x-\sqrt{2xy}-4y=0`
`<=> x+\sqrt{2xy}-2\sqrt{2xy}-4y=0`
`<=> \sqrt{x}*(\sqrt{x}+\sqrt{2y})-2\sqrt{2y}*(\sqrt{x}+\sqrt{2y})=0`
`<=> (\sqrt{x}+\sqrt{2y})(\sqrt{x}-2\sqrt{2y})=0`
`<=> \sqrt{x}=2\sqrt{2y}`(do `\sqrt{x}+\sqrt{2y} > 0` với mọi x, y > 0)
`<=> x=8y`
Khi đó: `P=(\root[3]{xy}(3\root[3]{x}-2\root[3]{y}))/(\sqrt{2xy})`
`=(\root[3]{8y*y}(3\root[3]{8y}-2\root[3]{y}))/(\sqrt{2*8y*y})`
`=(2\root[3]{y^2}(6\root[3]{y}-2\root[3]{y}))/(\sqrt{16y^2})`
`=(2\root[3]{y^2}*4\root[3]{y})/(4y)`
`=(8y)/(4y)=2`
#`Toru`
