Hãy thu gọn \(P=\frac{x}{(\sqrt{x}+\sqrt{y})(1-\sqrt{y})}-\frac{y}{(\sqrt{x}+\sqrt{y})(\sqrt{x}+1)}-\frac{x y}{(\sqrt{x}+1)(1-\sqrt{y})}\) ta

* Hướng dẫn giải

Điều kiện \(x \geq 0 ; y \geq 0 ; y \neq 1 ; x+y \neq 0\)

Ta có

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

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