Cho P= $\frac{\sqrt{x}}{\sqrt{xy}+\sqrt{x}+2}+$ $\frac{\sqrt{y}}{\sqrt{yz}+\sqrt{x}+1}+$ $\frac{2\sqrt{z}}{\sqrt{xz}+2\sqrt{z}+2}$
Biết xyz=4.Tính $\sqrt{P}$
Cho P= $\frac{\sqrt{x}}{\sqrt{xy}+\sqrt{x}+2}+$ $\frac{\sqrt{y}}{\sqrt{yz}+\sqrt{x}+1}+$ $\frac{2\sqrt{z}}{\sqrt{xz}+2\sqrt{z}+2}$
Biết xyz=4.Tính $\sqrt{P}$
`P=(\sqrt{x})/(\sqrt{xy}+\sqrt{x}+2)+(\sqrt{y})/(\sqrt{yz}+\sqrt{y}+1)+(2\sqrt{z})/(\sqrt{xz}+2\sqrt{z}+2)`
`P=(\sqrt{x}\sqrt{z})/(\sqrt{z}(\sqrt{xy}+\sqrt{x}+2))+(\sqrt{y}\sqrt{xz})/(\sqrt{xz}(\sqrt{yz}+\sqrt{y}+1))+(2\sqrt{z})/(\sqrt{xz}+2\sqrt{z}+2)`
`P=(\sqrt{xz})/(\sqrt{xyz}+\sqrt{xz}+2\sqrt{z})+(\sqrt{xyz})/(\sqrt{xyz^2}+\sqrt{xyz}+\sqrt{xz})+(2\sqrt{z})/(\sqrt{xz}+2\sqrt{z}+2)`
`P=(\sqrt{xz})/(\sqrt{4}+\sqrt{xz}+2\sqrt{z})+(\sqrt{4})/(\sqrt{4*z}+\sqrt{4}+\sqrt{xz})+(2\sqrt{z})/(\sqrt{xz}+2\sqrt{z}+2)`
`P=(\sqrt{xz})/(2+\sqrt{xz}+2\sqrt{z})+(2)/(2\sqrt{z}+2+\sqrt{xz})+(2\sqrt{z})/(\sqrt{xz}+2\sqrt{z}+2)`
`P=(\sqrt{xz}+2+2\sqrt{z})/(\sqrt{xz}+2+2\sqrt{z}`
`P=1`
`⇒\sqrt{P} = \sqrt{1} = 1`
Vậy `\sqrt{P} = 1`
Đáp án:
$\sqrt{P}=1$
Giải thích các bước giải:
$P=\dfrac{\sqrt{x}}{\sqrt{xy}+\sqrt{x}+2}+\dfrac{\sqrt{y}}{\sqrt{yz}+\sqrt{y}+1}+\dfrac{2\sqrt{z}}{\sqrt{xz}+2\sqrt{z}+2}$
$=\dfrac{\sqrt{xz}}{\sqrt{xyz}+\sqrt{xz}+2\sqrt{z}}+\dfrac{\sqrt{xyz}}{\sqrt{xyz^2}+\sqrt{xyz}+\sqrt{xz}}+\dfrac{2\sqrt{z}}{\sqrt{xz}+2\sqrt{z}+2}$
$=\dfrac{\sqrt{xz}}{\sqrt{4}+\sqrt{xz}+2\sqrt{z}}+\dfrac{\sqrt{4}}{\sqrt{4.z}+\sqrt{4}+\sqrt{xz}}+\dfrac{2\sqrt{z}}{\sqrt{xz}+2\sqrt{z}+2}$
$=\dfrac{\sqrt{xz}}{\sqrt{xz}+2+2\sqrt{z}}+\dfrac{2}{\sqrt{xz}+2+2\sqrt{z}}+\dfrac{2\sqrt{z}}{\sqrt{xz}+2+2\sqrt{z}}$
$=\dfrac{\sqrt{xz}+2+2\sqrt{z}}{\sqrt{xz}+2+2\sqrt{z}}$
$=1$
$⇒ \sqrt{P}=\sqrt{1}=1$