Ta có: \(\left(\frac{1}{\sqrt{x}}+\frac{1}{\sqrt{y}}\right)\cdot\frac{2}{\sqrt{x}+\sqrt{y}}+\frac{1}{x}+\frac{1}{y}\)
\(=\frac{\sqrt{x}+\sqrt{y}}{\sqrt{xy}}\cdot\frac{2}{\sqrt{x}+\sqrt{y}}+\frac{x+y}{xy}\)
\(=\frac{2}{\sqrt{xy}}+\frac{x+y}{xy}=\frac{x+2\sqrt{xy}+y}{xy}=\frac{\left(\sqrt{x}+\sqrt{y}\right)^2}{xy}\)
Ta có: \(\frac{\sqrt{x^3}+y\cdot\sqrt{x}+x\sqrt{y}+\sqrt{y^3}}{\sqrt{xy^3}+\sqrt{x^3y}}\)
\(=\frac{x\cdot\sqrt{x}+y\cdot\sqrt{x}+x\cdot\sqrt{y}+y\cdot\sqrt{y}}{\sqrt{xy}\left(x+y\right)}\)
\(=\frac{\sqrt{x}\left(x+y\right)+\sqrt{y}\left(x+y\right)}{\sqrt{xy}\left(x+y\right)}=\frac{\sqrt{x}+\sqrt{y}}{\sqrt{xy}}\)
Ta có: \(A=\left\lbrack\left(\frac{1}{\sqrt{x}}+\frac{1}{\sqrt{y}}\right)\cdot\frac{2}{\sqrt{x}+\sqrt{y}}+\frac{1}{x}+\frac{1}{y}\right\rbrack:\frac{\sqrt{x^3}+y\cdot\sqrt{x}+x\sqrt{y}+\sqrt{y^3}}{\sqrt{xy^3}+\sqrt{x^3y}}\)
\(=\frac{\left(\sqrt{x}+\sqrt{y}\right)^2}{xy}:\frac{\sqrt{x}+\sqrt{y}}{\sqrt{xy}}\)
\(=\frac{\left(\sqrt{x}+\sqrt{y}\right)^2}{xy}\cdot\frac{\sqrt{xy}}{\sqrt{x}+\sqrt{y}}=\frac{\sqrt{x}+\sqrt{y}}{\sqrt{xy}}\)
