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ta có:\(\frac{\left(x\sqrt{y}+y\sqrt{x}\right)\cdot\left(\sqrt{x}-\sqrt{y}\right)}{\sqrt{xy}}=\frac{\sqrt{xy}\left(\sqrt{x}+\sqrt{y}\right)\left(\sqrt{x}-\sqrt{y}\right)}{\sqrt{xy}}=x-y\)
vậy.....
\(\frac{\left(x\sqrt{y}+y\sqrt{x}\right).\left(\sqrt{x}-\sqrt{y}\right)}{\sqrt{xy}}\)
\(=\frac{\sqrt{xy}.\left(\sqrt{x}+\sqrt{y}\right)\left(\sqrt{x}-\sqrt{y}\right)}{\sqrt{xy}}\)
\(=\left(\sqrt{x}+\sqrt{y}\right)\left(\sqrt{x}-\sqrt{y}\right)\)
\(=x-y\)( đpcm )
a: \(=\dfrac{3}{2}\sqrt{6}+\dfrac{2}{3}\sqrt{6}-2\sqrt{3}=\dfrac{13}{6}\sqrt{6}-2\sqrt{3}\)
b: \(VT=\dfrac{\sqrt{xy}\left(\sqrt{x}+\sqrt{y}\right)}{\sqrt{xy}}\cdot\left(\sqrt{x}+\sqrt{y}\right)=\left(\sqrt{x}+\sqrt{y}\right)^2\)
c: \(VT=\dfrac{\sqrt{y}}{\sqrt{x}\left(\sqrt{x}-\sqrt{y}\right)}+\dfrac{\sqrt{x}}{\sqrt{y}\left(\sqrt{y}-\sqrt{x}\right)}\)
\(=\dfrac{y-x}{\sqrt{xy}\left(\sqrt{x}-\sqrt{y}\right)}=\dfrac{-\left(\sqrt{x}+\sqrt{y}\right)}{\sqrt{xy}}\)
Ta có : \(P=\frac{\frac{\left(x-y\right)^3}{\left(\sqrt{x}+\sqrt{y}\right)^3}+2x\sqrt{x}+y\sqrt{y}}{x\sqrt{x}+y\sqrt{y}}+\frac{3\left(\sqrt{xy}-y\right)}{x-y}\)
=> \(P=\frac{\frac{\left(\sqrt{x}+\sqrt{y}\right)^3\left(\sqrt{x}-\sqrt{y}\right)^3}{\left(\sqrt{x}+\sqrt{y}\right)^3}+2x\sqrt{x}+y\sqrt{y}}{\sqrt{x}^3+\sqrt{y}^3}+\frac{3\sqrt{y}\left(\sqrt{x}-\sqrt{y}\right)}{\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}+\sqrt{y}\right)}\)
=> \(P=\frac{\left(\sqrt{x}-\sqrt{y}\right)^3+2x\sqrt{x}+y\sqrt{y}}{\sqrt{x}^3+\sqrt{y}^3}+\frac{3\sqrt{y}}{\sqrt{x}+\sqrt{y}}\)
=> \(P=\frac{x\sqrt{x}-3x\sqrt{y}+3y\sqrt{x}-y\sqrt{y}+2x\sqrt{x}+y\sqrt{y}}{\left(x+y\right)\left(x-\sqrt{xy}+y\right)}+\frac{3\sqrt{y}}{\sqrt{x}+\sqrt{y}}\)
=> \(P=\frac{3x\sqrt{x}-3x\sqrt{y}+3y\sqrt{x}}{\left(x+y\right)\left(x-\sqrt{xy}+y\right)}+\frac{3\sqrt{y}}{\sqrt{x}+\sqrt{y}}\)
=> \(P=\frac{3\sqrt{x}\left(x-\sqrt{xy}+y\right)}{\left(\sqrt{x}+\sqrt{y}\right)\left(x-\sqrt{xy}+y\right)}+\frac{3\sqrt{y}}{\sqrt{x}+\sqrt{y}}\)
=> \(P=\frac{3\sqrt{x}}{\sqrt{x}+\sqrt{y}}+\frac{3\sqrt{y}}{\sqrt{x}+\sqrt{y}}\)
=> \(P=\frac{3\sqrt{x}+3\sqrt{y}}{\sqrt{x}+\sqrt{y}}=\frac{3\left(\sqrt{x}+\sqrt{y}\right)}{\sqrt{x}+\sqrt{y}}=3\)
\(A=\frac{x\sqrt{y}-y\sqrt{x}}{\sqrt{xy}}+\frac{x-y}{\sqrt{x}-\sqrt{y}}\)
\(A=\frac{\sqrt{x}\cdot\sqrt{x}\cdot\sqrt{y}-\sqrt{y}\cdot\sqrt{y}\cdot\sqrt{x}}{\sqrt{xy}}+\frac{\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}+\sqrt{y}\right)}{\sqrt{x}-\sqrt{y}}\)
\(A=\frac{\sqrt{xy}\left(\sqrt{x}-\sqrt{y}\right)}{\sqrt{xy}}+\sqrt{x}+\sqrt{y}\)
\(A=\sqrt{x}-\sqrt{y}+\sqrt{x}+\sqrt{y}\)
\(A=2\sqrt{x}\)
\(A=\left(\dfrac{4\sqrt{xy}+x-2\sqrt{xy}+y}{2\left(x-y\right)}\right)\cdot\dfrac{2\sqrt{x}}{\sqrt{x}+\sqrt{y}}-\dfrac{\sqrt{y}}{\sqrt{x}-\sqrt{y}}\)
\(=\dfrac{\left(\sqrt{x}+\sqrt{y}\right)^2}{\left(x-y\right)}\cdot\dfrac{\sqrt{x}}{\sqrt{x}+\sqrt{y}}-\dfrac{\sqrt{y}}{\sqrt{x}-\sqrt{y}}\)
\(=\dfrac{\sqrt{x}-\sqrt{y}}{\sqrt{x}-\sqrt{y}}=1\)
Ta có: \(B=\dfrac{x\sqrt{y}-y\sqrt{x}}{\sqrt{xy}}+\dfrac{x-y}{\sqrt{x}-\sqrt{y}}\)
\(\Leftrightarrow B=\dfrac{\sqrt{xy}\left(\sqrt{x}-\sqrt{y}\right)}{\sqrt{xy}}+\dfrac{\left(\sqrt{x}+\sqrt{y}\right)\left(\sqrt{x}-\sqrt{y}\right)}{\sqrt{x}-\sqrt{y}}\)
\(\Leftrightarrow B=\sqrt{x}-\sqrt{y}+\sqrt{x}+\sqrt{y}\)
\(\Leftrightarrow B=2\sqrt{x}\)
ĐK: \(x,y>0;x\ne y\)
\(VT=\frac{x\sqrt{y}-y\sqrt{x}}{\sqrt{x}-\sqrt{y}}\)
\(=\frac{\sqrt{x^2y}-\sqrt{xy^2}}{\sqrt{x}-\sqrt{y}}\)
\(=\frac{\sqrt{xy}\left(\sqrt{x}-\sqrt{y}\right)}{\sqrt{x}-\sqrt{y}}=\sqrt{xy}=VP\)
\(\Rightarrow\)đpcm
Ta có: \(\frac{x\sqrt{y}-y\sqrt{x}}{\sqrt{x}-\sqrt{y}}=\frac{\sqrt{xy}\left(\sqrt{x}-\sqrt{y}\right)}{\sqrt{x}-\sqrt{y}}=\sqrt{xy}\)
TK nha!