[tex]\bf \left[ \cfrac{(2a^{-3}b^4)^2}{(3a^5b)^{-2}} \right]^{-1}\implies \left[ \cfrac{(3a^5b)^{-2}} {(2a^{-3}b^4)^2}\right]\implies \cfrac{\frac{1}{(3a^5b)^{2}}} {(2a^{-3}b^4)^2}
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[tex]\bf \cfrac{\frac{1}{(3a^5b)^{2}}} {2^2a^{-6}b^8}\implies \cfrac{\frac{1}{(3a^5b)^{2}}} {2^2\frac{1}{a^6}b^8}\implies \cfrac{\frac{1}{3^2a^{10}b^2}}{\frac{4b^8}{a^6}}\implies \cfrac{1}{3^2a^{10}b^2}\cdot \cfrac{a^6}{4b^8}
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\cfrac{1}{3^24a^{10}b^2b^8}\cdot \cfrac{1}{a^{-6}}\implies \cfrac{1}{9\cdot 4a^{10}a^{-6}b^2b^8}
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\cfrac{1}{36a^{10-6}b^{2+8}}\implies \boxed{\cfrac{1}{36a^4b^{10}}}[/tex]
