i: \(\dfrac{2x}{x^2+2xy}+\dfrac{y}{xy-2y^2}+\dfrac{4}{x^2-4y^2}\)
\(=\dfrac{2x}{x\left(x+2y\right)}+\dfrac{y}{y\left(x-2y\right)}+\dfrac{4}{\left(x-2y\right)\left(x+2y\right)}\)
\(=\dfrac{2}{x+2y}+\dfrac{1}{x-2y}+\dfrac{4}{\left(x-2y\right)\left(x+2y\right)}\)
\(=\dfrac{2x-4y+x+2y+4}{\left(x+2y\right)\left(x-2y\right)}=\dfrac{3x+2y+4}{\left(x+2y\right)\left(x-2y\right)}\)
k: \(=\dfrac{1}{x}-\dfrac{1}{x+1}+\dfrac{1}{x+1}-\dfrac{1}{x+2}+...+\dfrac{1}{x+5}-\dfrac{1}{x+6}\)
=\(\dfrac{1}{x}-\dfrac{1}{x+6}\)
\(=\dfrac{x+6-x}{x\left(x+6\right)}=\dfrac{6}{x\left(x+6\right)}\)
m: \(=\dfrac{x^2+1}{2\left(x+1\right)}\cdot\dfrac{2xy}{2xy}=\dfrac{x^2+1}{2x+2}\)
n: \(=\dfrac{x+3}{2x}\cdot\dfrac{x+1}{x+1}=\dfrac{x+3}{2x}\)
