Simplify: $\;$ $\dfrac{1}{a^2} \sqrt{\left(a^6 + \dfrac{3 a^4}{b^{-2}} + \dfrac{a^2 b^4}{3^{-1}} + \dfrac{1}{b^{-6}}\right)^{\frac{2}{3}}} + \left[\dfrac{\left(b^{\frac{2}{3}} - a^{\frac{2}{3}}\right)^3 - 2a^2 - b^2}{a^2 + \left(b^{\frac{2}{3}} - a^{\frac{2}{3}}\right)^3 + 2 b^2}\right]^{-3}$
$\dfrac{1}{a^2} \sqrt{\left(a^6 + \dfrac{3 a^4}{b^{-2}} + \dfrac{a^2 b^4}{3^{-1}} + \dfrac{1}{b^{-6}}\right)^{\frac{2}{3}}} + \left[\dfrac{\left(b^{\frac{2}{3}} - a^{\frac{2}{3}}\right)^3 - 2a^2 - b^2}{a^2 + \left(b^{\frac{2}{3}} - a^{\frac{2}{3}}\right)^3 + 2 b^2}\right]^{-3}$ $\;\;\; \cdots \; (1)$
Consider the expression $\;\;$ $\dfrac{1}{a^2} \sqrt{\left(a^6 + \dfrac{3 a^4}{b^{-2}} + \dfrac{a^2 b^4}{3^{-1}} + \dfrac{1}{b^{-6}}\right)^{\frac{2}{3}}}$
$= \dfrac{1}{a^2} \left[\left(a^6 + 3 a^4 b^2 + 3 a^2 b^4 + b^6\right)^{\frac{2}{3}}\right]^{\frac{1}{2}}$
$= \dfrac{1}{a^2} \left[\left(a^2\right)^3 + 3 \left(a^2\right)^2 b^2 + 3 a^2 \left(b^2\right)^2 + \left(b^2\right)^3\right]^{\frac{1}{3}}$
$= \dfrac{1}{a^2} \left[\left(a^2 + b^2\right)^3\right]^{\frac{1}{3}}$
$= \dfrac{a^2 + b^2}{a^2}$
$= 1 + \dfrac{b^2}{a^2}$ $\;\;\; \cdots \; (2)$
Consider the expression $\;\;$ $\left[\dfrac{\left(b^{\frac{2}{3}} - a^{\frac{2}{3}}\right)^3 - 2a^2 - b^2}{a^2 + \left(b^{\frac{2}{3}} - a^{\frac{2}{3}}\right)^3 + 2 b^2}\right]^{-3}$
$= \left[\dfrac{\left(b^{\frac{2}{3}}\right)^3 - 3 \left(b^{\frac{2}{3}}\right)^2 a^{\frac{2}{3}} + 3 b^{\frac{2}{3}} \left(a^{\frac{2}{3}}\right)^2 - \left(a^{\frac{2}{3}}\right)^3 - 2a^2 - b^2}{a^2 + \left(b^{\frac{2}{3}}\right)^3 - 3 \left(b^{\frac{2}{3}}\right)^2 a^{\frac{2}{3}} + 3 b^{\frac{2}{3}} \left(a^{\frac{2}{3}}\right)^2 - \left(a^{\frac{2}{3}}\right)^3 + 2b^2} \right]^{-3}$
$= \left[\dfrac{b^2 - 3 b^{\frac{4}{3}} a^{\frac{2}{3}} + 3 b^{\frac{2}{3}} a^{\frac{4}{3}} - a^2 - 2a^2 - b^2}{a^2 + b^2 - 3 b^{\frac{4}{3}} a^{\frac{2}{3}} + 3 b^{\frac{2}{3}} a^{\frac{4}{3}} - a^2 + 2b^2}\right]^{-3}$
$= \left[\dfrac{3b^2 - 3 b^{\frac{4}{3}} a^{\frac{2}{3}} + 3 b^{\frac{2}{3}} a^{\frac{4}{3}}}{-3a^2 - 3b^{\frac{4}{3}} a^{\frac{2}{3}} + 3 b^{\frac{2}{3}} a^{\frac{4}{3}}}\right]^3$
$= \left[\dfrac{b^2 - a^{\frac{2}{3}} b^{\frac{2}{3}} \left(b^{\frac{2}{3}} - a^{\frac{2}{3}}\right)}{-a^2 + a^{\frac{2}{3}} b^{\frac{2}{3}} \left(a^{\frac{2}{3}} - b^{\frac{2}{3}}\right)}\right]^3$
$= \left[\dfrac{b^{\frac{2}{3}} \left(b^{\frac{4}{3}} - a^{\frac{2}{3}} b^{\frac{2}{3}} + a^{\frac{4}{3}}\right)}{-a^{\frac{2}{3}} \left(a^{\frac{4}{3}} - a^{\frac{2}{3}} b^{\frac{2}{3}} + b^{\frac{2}{3}}\right)}\right]^3$
$= \left[\dfrac{- b^{\frac{2}{3}}}{a^{\frac{2}{3}}}\right]^3$
$= \dfrac{- b^2}{a^2}$ $\;\;\; \cdots \; (3)$
In view of expressions $(2)$ and $(3)$, expression $(1)$ becomes
$1 + \dfrac{b^2}{a^2} - \dfrac{b^2}{a^2}$
$= 1$