Sửa đề: \(P=\left(\frac{\sqrt{x}+1}{\sqrt{xy}+1}+\frac{\sqrt{xy}+\sqrt{x}}{1-\sqrt{xy}}+1\right):\left(1-\frac{\sqrt{xy}+\sqrt{x}}{\sqrt{xy}-1}-\frac{\sqrt{x}+1}{\sqrt{xy}+1}\right)\)
a: \(\frac{\sqrt{x}+1}{\sqrt{xy}+1}+\frac{\sqrt{xy}+\sqrt{x}}{1-\sqrt{xy}}+1\)
\(=\frac{\left(\sqrt{x}+1\right)\left(\sqrt{xy}-1\right)-\left(\sqrt{xy}+\sqrt{x}\right)\left(\sqrt{xy}+1\right)+xy-1}{xy-1}\)
\(=\frac{x\cdot\sqrt{y}-\sqrt{x}+\sqrt{xy}-1-xy-\sqrt{xy}-x\sqrt{y}-\sqrt{x}+xy-1}{xy-1}\)
\(=\frac{-2\sqrt{x}-2}{xy-1}=-\frac{2\left(\sqrt{x}+1\right)}{xy-1}\)
Ta có: \(1-\frac{\sqrt{xy}+\sqrt{x}}{\sqrt{xy}-1}-\frac{\sqrt{x}+1}{\sqrt{xy}+1}\)
\(=\frac{xy-1-\left(\sqrt{xy}+\sqrt{x}\right)\left(\sqrt{xy}+1\right)-\left(\sqrt{x}+1\right)\left(\sqrt{xy}-1\right)}{xy-1}\)
\(=\frac{xy-1-xy-\sqrt{xy}-x\sqrt{y}-\sqrt{x}-x\sqrt{y}+\sqrt{x}-\sqrt{xy}+1}{xy-1}\)
\(=\frac{-2\sqrt{xy}-2x\sqrt{y}}{xy-1}=-\frac{2\sqrt{xy}\left(\sqrt{x}+1\right)}{xy-1}\)
Ta có: \(P=\left(\frac{\sqrt{x}+1}{\sqrt{xy}+1}+\frac{\sqrt{xy}+\sqrt{x}}{1-\sqrt{xy}}+1\right):\left(1-\frac{\sqrt{xy}+\sqrt{x}}{\sqrt{xy}-1}-\frac{\sqrt{x}+1}{\sqrt{xy}+1}\right)\)
\(=-\frac{2\left(\sqrt{x}+1\right)}{xy-1}:\frac{-2\sqrt{xy}\left(\sqrt{x}+1\right)}{xy-1}=\frac{2\left(\sqrt{x}+1\right)}{xy-1}\cdot\frac{xy-1}{2\sqrt{xy}\left(\sqrt{x}+1\right)}=\frac{1}{\sqrt{xy}}\)
b: \(x=\sqrt[3]{4-2\sqrt6}+\sqrt[3]{4+2\sqrt6}\)
=>\(x^3=4-2\sqrt6+4+2\sqrt6+3\cdot x\cdot\sqrt[3]{\left(4-2\sqrt6\right)\left(4+2\sqrt6\right)}\)
=>\(x^3=8+3x\cdot\sqrt[3]{16-24}=8+3x\cdot\left(-2\right)=8-6x\)
=>\(x^3+6x-8=0\)
\(y=x^2+6\)
=>xy=x^3+6x=8
=>\(P=\frac{1}{\sqrt8}=\frac{1}{2\sqrt2}=\frac{\sqrt2}{4}\)
