Simplify: $\;$ $2 \left(a + b\right)^{-1} \left(ab\right)^{\frac{1}{2}} \left[1 + \dfrac{1}{4} \left(\sqrt{\dfrac{a}{b}} - \sqrt{\dfrac{b}{a}}\right)^2\right]^{\frac{1}{2}}$ $\;\;$ for $a > 0, \; b > 0$
$2 \left(a + b\right)^{-1} \left(ab\right)^{\frac{1}{2}} \left[1 + \dfrac{1}{4} \left(\sqrt{\dfrac{a}{b}} - \sqrt{\dfrac{b}{a}}\right)^2\right]^{\frac{1}{2}}$
$= \dfrac{2}{a + b} \times \sqrt{ab} \times \left[1 + \dfrac{1}{4} \left(\dfrac{a}{b} - 2 \sqrt{\dfrac{a}{b}} \times \sqrt{\dfrac{b}{a}} + \dfrac{b}{a}\right) \right]^{\frac{1}{2}}$
$= \dfrac{2}{a + b} \times \sqrt{ab} \times \left[1 + \dfrac{a}{4b} - \dfrac{1}{2} + \dfrac{b}{4a}\right]^{\frac{1}{2}}$
$= \dfrac{2}{a + b} \times \sqrt{ab} \times \left[\dfrac{1}{2} + \dfrac{a}{4b} + \dfrac{b}{4a}\right]^{\frac{1}{2}}$
$= \dfrac{2}{a + b} \times \sqrt{ab} \times \left[\dfrac{2ab + a^2 + b^2}{4ab}\right]^{\frac{1}{2}}$
$= \dfrac{\left[\left(a + b\right)^2\right]^{\frac{1}{2}}}{a + b}$
$= \dfrac{a + b}{a + b}$
$= 1$ $\;\;\;$ [when $a > 0, \; b > 0 \;$ then $\; a + b > 0$]