Simplify the following and express with positive index :

$(32)^{-\dfrac{2}{5}} ÷ (125)^{-\dfrac{2}{3}}$

Simplifying the expression :

$\Rightarrow (32)^{-\dfrac{2}{5}} ÷ (125)^{-\dfrac{2}{3}} = (2^5)^{-\dfrac{2}{5}} ÷ (5^3)^{-\dfrac{2}{3}} \\[1em] = (2)^{5 \times -\dfrac{2}{5}} ÷ (5)^{3 \times -\dfrac{2}{3}} \\[1em] = (2)^{-2} ÷ 5^{-2} \\[1em] = \dfrac{1}{2^2} ÷ \dfrac{1}{5^2} \\[1em] = \dfrac{1}{2^2} \times 5^2 \\[1em] = \dfrac{1}{4} \times 25 \\[1em] = \dfrac{25}{4} = 6\dfrac{1}{4}.$

Hence, $(32)^{-\dfrac{2}{5}} ÷ (125)^{-\dfrac{2}{3}} = 6\dfrac{1}{4}$.