\(\dfrac{a\dfrac{1}{b}}{b\dfrac{1}{a}}=\dfrac{a+\dfrac{1}{b}}{b+\dfrac{1}{a}}=\dfrac{\dfrac{ab}{b}+\dfrac{1}{b}}{\dfrac{ab}{a}+\dfrac{1}{a}}=\dfrac{\dfrac{ab+1}{b}}{\dfrac{ab+1}{a}}=\dfrac{ab+1}{b}\cdot\dfrac{a}{ab+1}=\dfrac{a}{b}\)

So \(\dfrac{a\dfrac{1}{b}}{b\dfrac{1}{a}}=\dfrac{a}{b}\)

Easy ^.^