a: \(\dfrac{x^3}{x^2+1}+\dfrac{x}{x^2+1}\)
\(=\dfrac{x^3+x}{x^2+1}\)
\(=\dfrac{x\left(x^2+1\right)}{x^2+1}=x\)
b:
ĐKXĐ: \(x\notin\left\{1;-1\right\}\)
\(\dfrac{x+1}{2x-2}+\dfrac{-2x}{x^2-1}\)
\(=\dfrac{x+1}{2\left(x-1\right)}-\dfrac{2x}{\left(x-1\right)\left(x+1\right)}\)
\(=\dfrac{\left(x+1\right)^2-4x}{2\left(x-1\right)\left(x+1\right)}=\dfrac{\left(x-1\right)^2}{2\left(x-1\right)\left(x+1\right)}=\dfrac{x-1}{2\left(x-1\right)}\)
c: ĐKXĐ: \(\left\{{}\begin{matrix}x\ne0;y\ne0\\x\ne y\end{matrix}\right.\)
\(\dfrac{1}{xy-x^2}-\dfrac{1}{y^2-xy}\)
\(=\dfrac{1}{x\left(y-x\right)}-\dfrac{1}{y\left(y-x\right)}\)
\(=\dfrac{y-x}{xy\left(y-x\right)}=\dfrac{1}{xy}\)
d: ĐKXĐ: \(x\notin\left\{2;-2\right\}\)
\(\dfrac{5x+10}{4x-8}\cdot\dfrac{4-2x}{x+2}\)
\(=\dfrac{5\left(x+2\right)}{4\left(x-2\right)}\cdot\dfrac{-2\left(x-2\right)}{x+2}\)
\(=\dfrac{5\cdot\left(-2\right)}{4}=-\dfrac{10}{4}=-\dfrac{5}{2}\)
a.
\(\dfrac{x^3}{x^2+1}+\dfrac{x}{x^2+1}=\dfrac{x^3+x}{x^2+1}=\dfrac{x\left(x^2+1\right)}{x^2+1}=x\)
b.
\(\dfrac{x+1}{2x-2}+\dfrac{-2x}{x^2-1}=\dfrac{x+1}{2\left(x-1\right)}-\dfrac{2x}{\left(x-1\right)\left(x+1\right)}=\dfrac{\left(x+1\right)^2}{2\left(x-1\right)\left(x+1\right)}-\dfrac{4x}{2\left(x-1\right)\left(x+1\right)}\)
\(=\dfrac{x^2+2x+1-4x}{2\left(x-1\right)\left(x+1\right)}=\dfrac{\left(x-1\right)^2}{2\left(x-1\right)\left(x+1\right)}=\dfrac{x-1}{2\left(x+1\right)}\)
c.
\(\dfrac{1}{xy-x^2}-\dfrac{1}{y^2-xy}=\dfrac{1}{x\left(y-x\right)}-\dfrac{1}{y\left(y-x\right)}=\dfrac{y}{xy\left(y-x\right)}-\dfrac{x}{xy\left(y-x\right)}\)
\(=\dfrac{y-x}{xy\left(y-x\right)}=\dfrac{1}{xy}\)
d.
\(\dfrac{5x+10}{4x-8}.\dfrac{4-2x}{x+2}=\dfrac{5\left(x+2\right)}{2\left(x-2\right)}.\dfrac{-2\left(x-2\right)}{x+2}=-5\)
