Question

[NeyuQ Nguyễn]

Rút gọn P

P = \(\left(\dfrac{\sqrt{x}+1}{\sqrt{xy}+1}+\dfrac{\sqrt{xy}+\sqrt{x}}{1-\sqrt{xy}}+1\right):\left(1-\dfrac{\sqrt{xy}+\sqrt{x}}{\sqrt{xy}-1}-\dfrac{\sqrt{x}+1}{\sqrt{xy}+1}\right)\)

Answer 1

[Nguyễn Lê Phước Thịnh]

ĐKXĐ: \(\left\{{}\begin{matrix}x>=0;y>=0\\x^2+y^2\ne1^2+1^2=2\end{matrix}\right.\)

\(\dfrac{\sqrt{x}+1}{\sqrt{xy}+1}+\dfrac{\sqrt{xy}+\sqrt{x}}{1-\sqrt{xy}}+1\)

\(=\dfrac{\sqrt{x}+1}{\sqrt{xy}+1}-\dfrac{\sqrt{xy}+\sqrt{x}}{\sqrt{xy}-1}+1\)

\(=\dfrac{\left(\sqrt{x}+1\right)\left(\sqrt{xy}-1\right)-\left(\sqrt{xy}+1\right)\left(\sqrt{xy}+\sqrt{x}\right)+xy-1}{xy-1}\)

\(=\dfrac{x\sqrt{y}-\sqrt{x}+\sqrt{xy}-1-xy-x\sqrt{y}-\sqrt{xy}-\sqrt{x}+xy-1}{xy-1}\)

\(=\dfrac{-2\sqrt{x}-2}{xy-1}\)

\(1-\dfrac{\sqrt{xy}+\sqrt{x}}{\sqrt{xy}-1}-\dfrac{\sqrt{x}+1}{\sqrt[]{xy}+1}\)

\(=\dfrac{xy-1-\left(\sqrt{xy}+\sqrt{x}\right)\left(\sqrt{xy}+1\right)-\left(\sqrt{x}+1\right)\left(\sqrt{xy}-1\right)}{xy-1}\)

\(=\dfrac{xy-1-xy-\sqrt{xy}-x\sqrt{y}-\sqrt{x}-x\sqrt{y}+\sqrt{x}-\sqrt{xy}+1}{xy-1}\)

\(=\dfrac{-2\sqrt{xy}-2x\sqrt{y}}{xy-1}\)

\(P=\left(\dfrac{\sqrt{x}+1}{\sqrt{xy}+1}+\dfrac{\sqrt{xy}+\sqrt{x}}{1-\sqrt{xy}}+1\right):\left(1-\dfrac{\sqrt{xy}+\sqrt{x}}{\sqrt{xy}-1}-\dfrac{\sqrt{x}+1}{\sqrt[]{xy}+1}\right)\)

\(=\dfrac{-2\left(\sqrt{x}+1\right)}{xy-1}:\dfrac{-2\sqrt{xy}\left(\sqrt{x}+1\right)}{xy-1}\)

\(=\dfrac{-2\left(\sqrt{x}+1\right)}{xy-1}\cdot\dfrac{xy-1}{-2\sqrt{xy}\left(\sqrt{x}+1\right)}=\dfrac{1}{\sqrt{xy}}\)