Answer
$a.) \dfrac{a^6}{b^3 c^9}$
$b.) \dfrac{a^8 c^6}{b^4}$
$c.) \dfrac{16b^6}{a^6}$
Work Step by Step
$a.) \ ( a^{-2} b c^3)^{-3} = a^{(-2)\times (-3)} b^{-3} c^{3\times (-3)} = a^6 b^{-3} c^{-9} = \dfrac{a^6}{b^3 c^9} $
$b.) \ \bigg(\dfrac{ a^{-4} b^2 }{c^3}\bigg)^{-2} = \dfrac{ a^{(-4)\times (-2)} b^{2\times (-2)} }{c^{3\times (-2)}} = \dfrac{a^8 b^{-4}}{c^{-6}} = \dfrac{a^8 c^6}{b^4}$
$c.) \ \bigg(\dfrac{3a^8b^2}{12a^5b^5}\bigg)^{-2} = \bigg(\dfrac{a^8a^{-5}b^2b^{-5}}{4}\bigg)^{-2} = \bigg(\dfrac{a^{8-5}b^{2-5}}{4}\bigg)^{-2} = \bigg(\dfrac{a^{3}b^{-3}}{4}\bigg)^{-2} = \dfrac{a^{3\times (-2)}b^{(-3)\times (-2)}}{4^{-2}} = 4^2a^{-6}b^6 = \dfrac{16b^6}{a^6}$
