Rút gọn biếu thức \(A=\left...

A

- Hướng dẫn giải

 \(\begin{array}{l} A=\left(\frac{\sqrt{x}+\sqrt{y}}{\sqrt{x} \sqrt{y}} \frac{2}{\sqrt{x}+\sqrt{y}}+\frac{x+y}{x y}\right): \frac{\left(\sqrt{x}^{3}+\sqrt{y}^{3}\right)+(y \sqrt{x}+x \sqrt{y})}{\sqrt{x y}(x+y)} \\ =\left(\frac{2}{\sqrt{x} \sqrt{y}+\frac{x+y}{x y}}\right): \frac{(\sqrt{x}+\sqrt{y})(x-\sqrt{x y}+y)+\sqrt{x y}(\sqrt{x}+\sqrt{y})}{\sqrt{x y}(x+y)} \end{array}\)

\(\begin{array}{l} =\frac{2 \sqrt{x y}+x+y}{x y}: \frac{(\sqrt{x}+\sqrt{y})(x-\sqrt{x y}+y+\sqrt{x y})}{\sqrt{x y}(x+y)} \\ =\frac{(\sqrt{x}+\sqrt{y})^{2}}{x y}: \frac{(\sqrt{x}+\sqrt{y})(x+y)}{\sqrt{x y}(x+y)} \\ =\frac{(\sqrt{x}+\sqrt{y})^{2}}{x y}: \frac{\sqrt{x}+\sqrt{y}}{\sqrt{x y}} \end{array}\)

\(\begin{array}{l} =\frac{(\sqrt{x}+\sqrt{y})^{2}}{x y} \frac{\sqrt{x y}}{\sqrt{x}+\sqrt{y}} \\ =\frac{\sqrt{x}+\sqrt{y}}{\sqrt{x y}} \\ =\frac{1}{\sqrt{x}}+\frac{1}{\sqrt{y}} \end{array}\)