$\text{a)}x\ne 0\text{i}y\ne 0\text{i}2x^2{y}^0\ne 0\text{czyli}\underline{\underline{x\ne 0,y\ne 0}}$

 

${\left(\frac{1}{2}{x}^{-2}{y}^{-1}\right)}^3:{\left(2x^2{y}^0\right)}^{-3}=\frac{{\left(\frac{1}{2}{x}^{-2}{y}^{-1}\right)}^3}{{\left(2x^2{y}^0\right)}^{-3}}=\frac{{\left(\frac{1}{2}\right)}^3\cdot {\left(x^{-2}\right)}^3\cdot {\left(y^{-1}\right)}^3}{2^{-3}\cdot {\left(x^2\right)}^{-3}}=$

$=\frac{\frac{1}{8}\cdot x^{-6}\cdot y^{-3}}{\frac{1}{8}\cdot x^{-6}}=y^{-3}=\frac{1}{y^3}$  

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$\text{b)}a\ne 0\text{i}b\ne 0\text{i}c\ne 0\text{i}\frac{a^2}{bc}\ne 0\Rightarrow \underline{\underline{a,b,c\ne 0}}$

 

$\left(a^{-1}{b}^{-2}{c}^{-3}\right)\cdot {\left(\frac{a^2}{bc}\right)}^{-1}=\frac{1}{a}\cdot \frac{1}{b^2}\cdot \frac{1}{c^3}\cdot \frac{bc}{a^2}=\frac{bc}{a^3{b}^2{c}^3}=\frac{1}{a^{3}b{c}^2}$

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$\text{c)}c\ne 0\text{i}{a}^6{c}^{-6}\ne 0\Rightarrow \underline{\underline{a,c\ne 0}}$

 

$\frac{{\left({\left(3ab\right)}^3{c}^{-2}\right)}^2}{a^6{c}^{-6}}=\frac{{\left(3ab\right)}^6{c}^{-4}}{a^6{c}^{-6}}=\frac{729a^6{b}^6{c}^{-4}}{a^6{c}^{-6}}=\frac{729b^6{c}^{-4}}{c^{-6}}=729b^6{c}^2$