Tính giá trị biểu thức: $P = \sqrt{(1 + x)(1 + y)(1 + z)}.(\dfrac{\sqrt{x}}{1 + x} + \dfrac{\sqrt{y}}{1 + y} + \dfrac{\sqrt{z}}{1 + z})$
Giải
Ta có: $x + 1 = x + \sqrt{xy} + \sqrt{xz} + \sqrt{yz} = \sqrt{x}(\sqrt{x} + \sqrt{y}) + \sqrt{z}(\sqrt{x} + \sqrt{y})$
$= (\sqrt{x} + \sqrt{y})(\sqrt{x} + \sqrt{z})$
Tương tự:
$y + 1 = (\sqrt{y} + \sqrt{x})(\sqrt{y} + \sqrt{z})$
$z + 1 = (\sqrt{z} + \sqrt{x})(\sqrt{z} + \sqrt{y})$
Do đó:
$P = \sqrt{(1 + x)(1 + y)(1 + z)}.(\dfrac{\sqrt{x}}{1 + x} + \dfrac{\sqrt{y}}{1 + y} + \dfrac{\sqrt{z}}{1 + z})$
$P = \sqrt{[(\sqrt{x} + \sqrt{y})(\sqrt{x} + \sqrt{z})(\sqrt{y} + \sqrt{z})]^2}.(\dfrac{\sqrt{x}}{(\sqrt{x} + \sqrt{y})(\sqrt{x} + \sqrt{z})} + \dfrac{\sqrt{y}}{(\sqrt{y} + \sqrt{x})(\sqrt{y} + \sqrt{z})} + \dfrac{\sqrt{z}}{(\sqrt{z} + \sqrt{x})(\sqrt{z} + \sqrt{y})})$
$P = (\sqrt{x} + \sqrt{y})(\sqrt{x} + \sqrt{z})(\sqrt{y} + \sqrt{z})[\dfrac{\sqrt{x}}{(\sqrt{x} + \sqrt{y})(\sqrt{x} + \sqrt{z})} + \dfrac{\sqrt{y}}{(\sqrt{y} + \sqrt{x})(\sqrt{y} + \sqrt{z})} + \dfrac{\sqrt{z}}{(\sqrt{z} + \sqrt{x})(\sqrt{z} + \sqrt{y})}]$
$P = \sqrt{x}(\sqrt{y} + \sqrt{z}) + \sqrt{y}(\sqrt{x} + \sqrt{z}) + \sqrt{z}(\sqrt{x} + \sqrt{y})$
$P = 2(\sqrt{xy} + \sqrt{zx} + \sqrt{yz}) = 2$