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\(\frac{x+\left(\sqrt{x}-\sqrt{z}\right)^2}{y+\left(\sqrt{y}-\sqrt{z}\right)^2}=\frac{\left(\sqrt{x}+\sqrt{y}-\sqrt{z}\right)^2-y+\left(\sqrt{x}-\sqrt{z}\right)^2}{\left(\sqrt{x}+\sqrt{y}-\sqrt{z}\right)^2-x+\left(\sqrt{y}-\sqrt{z}\right)^2}\)
\(=\frac{\left(\sqrt{x}+2\sqrt{y}-\sqrt{z}\right)\left(\sqrt{x}-\sqrt{z}\right)+\left(\sqrt{x}-\sqrt{z}\right)^2}{\left(2\sqrt{x}+\sqrt{y}-\sqrt{z}\right)\left(\sqrt{y}-\sqrt{z}\right)+\left(\sqrt{y}-\sqrt{z}\right)^2}\)
\(=\frac{\left(\sqrt{x}-\sqrt{z}\right)\left(2\sqrt{x}+2\sqrt{y}-2\sqrt{z}\right)}{\left(\sqrt{y}-\sqrt{z}\right)\left(2\sqrt{x}+2\sqrt{y}-2\sqrt{z}\right)}\)
\(=\frac{\sqrt{x}-\sqrt{z}}{\sqrt{y}-\sqrt{z}}\)
\(ĐKXĐ:x,y,z\ge0;x\ne y\ne z\)
Ta có :
\(\frac{x}{\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}-\sqrt{z}\right)}+\frac{y}{\left(\sqrt{y}-\sqrt{z}\right)\left(\sqrt{y}-\sqrt{x}\right)}+\frac{z}{\left(\sqrt{z}-\sqrt{x}\right)\left(\sqrt{z}-\sqrt{y}\right)}\)
\(=\frac{-x}{\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{z}-\sqrt{x}\right)}-\frac{y}{\left(\sqrt{y}-\sqrt{z}\right)\left(\sqrt{x}-\sqrt{y}\right)}-\frac{z}{\left(\sqrt{x}-\sqrt{z}\right)\left(\sqrt{z}-\sqrt{y}\right)}\)
\(=\frac{-x.\left(\sqrt{y}-\sqrt{z}\right)-y.\left(\sqrt{z}-\sqrt{x}\right)-z.\left(\sqrt{x}-\sqrt{y}\right)}{\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{y}-\sqrt{z}\right)\left(\sqrt{z}-\sqrt{x}\right)}\)
Xét \(-x.\left(\sqrt{y}-\sqrt{z}\right)-y.\left(\sqrt{z}-\sqrt{x}\right)-z.\left(\sqrt{x}-\sqrt{y}\right)\)
\(=-x\left(\sqrt{y}-\sqrt{z}\right)-y\sqrt{z}+y\sqrt{x}-z\sqrt{x}+z\sqrt{y}\)
\(=-x\left(\sqrt{y}-\sqrt{z}\right)+\sqrt{zx}\left(\sqrt{y}-\sqrt{z}\right)-\sqrt{yz}\left(\sqrt{y}-\sqrt{z}\right)+\sqrt{xy}\left(\sqrt{y}-\sqrt{z}\right)\)
\(=\left(\sqrt{y}-\sqrt{z}\right).\left(-x+\sqrt{zx}-\sqrt{zy}+\sqrt{xy}\right)\)
\(=\left(\sqrt{y}-\sqrt{z}\right).\left[\sqrt{x}.\left(\sqrt{z}-\sqrt{x}\right)-\sqrt{y}.\left(\sqrt{z}-\sqrt{x}\right)\right]\)
\(=\left(\sqrt{y}-\sqrt{z}\right)\left(\sqrt{z}-\sqrt{x}\right)\left(\sqrt{x}-\sqrt{y}\right)\)
\(=\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{y}-\sqrt{z}\right)\left(\sqrt{z}-\sqrt{x}\right)\)
Khi đó :
\(\frac{x}{\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}-\sqrt{z}\right)}+\frac{y}{\left(\sqrt{y}-\sqrt{z}\right)\left(\sqrt{y}-\sqrt{x}\right)}+\frac{z}{\left(\sqrt{z}-\sqrt{x}\right)\left(\sqrt{z}-\sqrt{y}\right)}\)
\(=\frac{\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{y}-\sqrt{z}\right)\left(\sqrt{z}-\sqrt{x}\right)}{\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{y}-\sqrt{z}\right)\left(\sqrt{z}-\sqrt{x}\right)}=1\)
Vậy \(\frac{x}{\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}-\sqrt{z}\right)}+\frac{y}{\left(\sqrt{y}-\sqrt{z}\right)\left(\sqrt{y}-\sqrt{x}\right)}+\frac{z}{\left(\sqrt{z}-\sqrt{x}\right)\left(\sqrt{z}-\sqrt{y}\right)}=1\)
mấy bài này làm hại não lắm :((
\(\frac{x}{\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}-\sqrt{z}\right)}+\frac{y}{\left(\sqrt{y}-\sqrt{z}\right)\left(\sqrt{y}-\sqrt{x}\right)}+\frac{z}{\left(\sqrt{z}-\sqrt{x}\right)\left(\sqrt{z}-\sqrt{y}\right)}\)
\(=\frac{-x}{\left(\sqrt{y}-\sqrt{x}\right)\left(\sqrt{x}-\sqrt{z}\right)}+\frac{-y}{\left(\sqrt{z}-\sqrt{y}\right)\left(\sqrt{y}-\sqrt{x}\right)}+\frac{-z}{\left(\sqrt{z}-\sqrt{y}\right)\left(\sqrt{x}-\sqrt{z}\right)}\)
\(=-\left[\frac{x\left(\sqrt{z}-\sqrt{y}\right)+y\left(\sqrt{x}-\sqrt{z}\right)+z\left(\sqrt{y}-\sqrt{x}\right)}{\left(\sqrt{x}-\sqrt{z}\right)\left(\sqrt{y}-\sqrt{x}\right)\left(\sqrt{z}-\sqrt{y}\right)}\right]\)
đến đây nhân tung ra rồi ghép cặp là okey nhé