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13 tháng 7 2015
\(\frac{x-y}{\sqrt{x}-\sqrt{y}}-\frac{\sqrt{x^3}-\sqrt{y^3}}{x-y}=\frac{\left(x-y\right)\left(\sqrt{x}+\sqrt{y}\right)-x\sqrt{x}+y\sqrt{y}}{\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}+\sqrt{y}\right)}=\frac{x\sqrt{x}+x\sqrt{y}-y\sqrt{x}-y\sqrt{y}-x\sqrt{x}+y\sqrt{y}}{\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}+\sqrt{y}\right)}\)
\(=\frac{x\sqrt{y}-y\sqrt{x}}{\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}+\sqrt{y}\right)}=\frac{\sqrt{x}\sqrt{y}\left(\sqrt{x}-\sqrt{y}\right)}{\left(\sqrt{x-\sqrt{y}}\right)\left(\sqrt{x}+\sqrt{y}\right)}=\frac{\sqrt{xy}}{\sqrt{x}+\sqrt{y}}\)
TH
30 tháng 9 2015
\(=\frac{2}{\sqrt{x}+\sqrt{y}}+\frac{1}{\sqrt{x}-\sqrt{y}}-\frac{3\sqrt{x}}{\left(\sqrt{x}+\sqrt{y}\right)\left(\sqrt{x}-\sqrt{y}\right)}\)
\(=\frac{2\left(\sqrt{x}-\sqrt{y}\right)+\sqrt{x}+\sqrt{y}-3\sqrt{x}}{\left(\sqrt{x}+\sqrt{y}\right)\left(\sqrt{x}-\sqrt{y}\right)}\)
\(=\frac{2\sqrt{x}-2\sqrt{y}+\sqrt{x}+\sqrt{y}-3\sqrt{x}}{\left(\sqrt{x}+\sqrt{y}\right)\left(\sqrt{x}-\sqrt{y}\right)}\)
\(=\frac{-\sqrt{y}}{x-y}\)
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ML
8 tháng 8 2015
\(=\sqrt{\left(\frac{x}{y}\right)^2+\left(\frac{y}{x}\right)^2-2.\frac{x}{y}.\frac{y}{x}}-\sqrt{\left(\frac{x}{y}\right)^2+\left(\frac{y}{x}\right)^2+2.\frac{x}{y}.\frac{y}{x}}\)
\(=\sqrt{\left(\frac{x}{y}-\frac{y}{x}\right)^2}-\sqrt{\left(\frac{x}{y}+\frac{y}{x}\right)^2}\)
\(=\left|\frac{x}{y}-\frac{y}{x}\right|-\left|\frac{x}{y}+\frac{y}{x}\right|\)
14 tháng 10 2018
\(\frac{x}{1+x^2}+\frac{2y}{1+y^2}+\frac{3z}{1+z^2}\)
\(=xyz.\left [ \frac{1}{yz(1+x^2)}+\frac{2}{xz(1+y^2)}+\frac{3}{xy(1+z^2)} \right ]\)
\(=xyz.\left [ \frac{1}{yz+x(x+y+z)}+\frac{2}{xz+y(x+y+z)}+\frac{3}{xy+z(x+y+z)} \right ]\)
\(=xyz.\left [ \frac{1}{(x+y)(x+z)}+\frac{2}{(x+y)(y+z)}+\frac{3}{(x+z)(y+z)} \right ]\)
\(=xyz.\frac{y+z+2(z+x)+3(x+y)}{(x+y)(y+z)(z+x)}=\frac{xyz(5x+4y+3z)}{(x+y)(y+z)(z+x)}\)
17 tháng 5 2015
\(\frac{x^3-x^2y-xy^2+y^3}{x^3+x^2y-xy^2-y^3}=\frac{\left(x^3-xy^2\right)-\left(x^2.y-y^3\right)}{\left(x^3-xy^2\right)+\left(x^2y-y^3\right)}=\frac{x.\left(x^2-y^2\right)-y.\left(x^2-y^2\right)}{x.\left(x^2-y^2\right)+y.\left(x^2-y^2\right)}=\frac{\left(x-y\right)\left(x^2-y^2\right)}{\left(x+y\right)\left(x^2-y^2\right)}=\frac{x-y}{x+y}\)
\(x\sqrt{x}+y\sqrt{y}+x-y\)
\(=\left(x\sqrt{x}+y\sqrt{y}\right)+\left(x-y\right)\)
\(=\left(\sqrt{x}+\sqrt{y}\right)\left(x-\sqrt{xy}+y\right)+\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}+\sqrt{y}\right)\)
\(=\left(\sqrt{x}+\sqrt{y}\right)\left(x-\sqrt{xy}+y+\sqrt{x}-\sqrt{y}\right)\)
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