a: \(A=\left[\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{2}{a+b}\cdot\left(\dfrac{1}{a}+\dfrac{1}{b}\right)\right]\cdot\dfrac{ab}{\left(a+b\right)^2}\)
\(=\left[\dfrac{b^2+a^2}{a^2b^2}+\dfrac{2}{a+b}\cdot\dfrac{a+b}{ab}\right]\cdot\dfrac{ab}{\left(a+b\right)^2}\)
\(=\left[\dfrac{b^2+a^2}{a^2b^2}+\dfrac{2}{ab}\right]\cdot\dfrac{ab}{\left(a+b\right)^2}\)
\(=\dfrac{a^2+2ab+b^2}{a^2b^2}\cdot\dfrac{ab}{\left(a+b\right)^2}\)
\(=\dfrac{\left(a+b\right)^2}{\left(a+b\right)^2}\cdot\dfrac{1}{ab}=\dfrac{1}{ab}\)
b: \(B=\left[\dfrac{1}{\left(2x-y\right)^2}+\dfrac{2}{4x^2-y^2}+\dfrac{1}{\left(2x+y\right)^2}\right]\cdot\dfrac{4x^2+4xy+y^2}{16x}\)
\(=\dfrac{\left(2x+y\right)^2+2\left(4x^2-y^2\right)+\left(2x-y\right)^2}{\left(2x-y\right)^2\cdot\left(2x+y\right)^2}\cdot\dfrac{\left(2x+y\right)^2}{16x}\)
\(=\dfrac{\left(2x+y\right)^2+2\cdot\left(2x+y\right)\left(2x-y\right)+\left(2x-y\right)^2}{\left(2x-y\right)^2}\cdot\dfrac{1}{16x}\)
\(=\dfrac{\left(2x+y+2x-y\right)^2}{\left(2x-y\right)^2}\cdot\dfrac{1}{16x}=\dfrac{16x^2}{16x\left(2x-y\right)^2}=\dfrac{x}{\left(2x-y\right)^2}\)
