a: ĐKXĐ: \(x\notin\left\{1;-1\right\}\)
\(\dfrac{x+1}{2x-2}+\dfrac{2}{1-x^2}+\dfrac{x-1}{2x+2}\)
\(=\dfrac{x+1}{2\left(x-1\right)}-\dfrac{2}{\left(x-1\right)\cdot\left(x+1\right)}+\dfrac{x-1}{2\left(x+1\right)}\)
\(=\dfrac{\left(x+1\right)^2-4+\left(x-1\right)^2}{2\left(x-1\right)\left(x+1\right)}\)
\(=\dfrac{x^2+2x+1-4+x^2-2x+1}{2\left(x-1\right)\left(x+1\right)}\)
\(=\dfrac{2x^2-2}{2\left(x-1\right)\left(x+1\right)}=\dfrac{2\left(x^2-1\right)}{2\left(x^2-1\right)}=1\)
b: ĐKXĐ: \(\left\{{}\begin{matrix}y\ne0\\x\ne\pm y\end{matrix}\right.\)
\(\left(x-y\right)\cdot\dfrac{\left(x+y\right)^2}{x^2y-y^3}\)
\(=\left(x-y\right)\cdot\dfrac{\left(x+y\right)^2}{y\left(x^2-y^2\right)}\)
\(=\left(x-y\right)\cdot\dfrac{\left(x+y\right)^2}{y\left(x-y\right)\left(x+y\right)}\)
\(=\dfrac{x+y}{y}\)
c: ĐKXĐ: \(\left\{{}\begin{matrix}x\ne0\\y\ne0\\x\ne y\end{matrix}\right.\)
\(\dfrac{1}{xy-x^2}-\dfrac{1}{y^2-xy}\)
\(=\dfrac{-1}{x\left(x-y\right)}+\dfrac{1}{y\left(x-y\right)}\)
\(=\dfrac{-y+x}{xy\left(x-y\right)}=\dfrac{1}{xy}\)
d: ĐKXĐ: \(x\notin\left\{-4;2\right\}\)
\(\dfrac{x^2-4}{3x+12}:\dfrac{2x-4}{x+4}\)
\(=\dfrac{\left(x-2\right)\left(x+2\right)}{3\left(x+4\right)}\cdot\dfrac{x+4}{2\left(x-2\right)}\)
\(=\dfrac{x+2}{6}\)
