a)

${\left(2x\right)}^2=2^2\cdot x^2=4x^2$

${\left(0,2x{y}^2\right)}^2={\left(0,2\right)}^2\cdot x^2\cdot {\left(y^2\right)}^2$ $=0,04x^2{y}^4$

${\left(\frac{3}{4}{a}^2\right)}^2={\left(\frac{3}{4}\right)}^2\cdot {\left(a^2\right)}^2$ ${\left(\frac{3}{4}{a}^2\right)}^2={\left(\frac{3}{4}\right)}^2\cdot {\left(a^2\right)}^2$ $=\frac{9}{16}{a}^4$

b)

${\left(3a{b}^3\right)}^3=3^3\cdot a^3\cdot {\left(b^3\right)}^3$ $=27a^3{b}^9$

${\left(-\frac{1}{2}{a}^6\right)}^3={\left(-\frac{1}{2}\right)}^3\cdot {\left(a^2\right)}^3$ $=-\frac{1}{8}{a}^6$

${\left(\frac{1}{4}x\right)}^3={\left(\frac{1}{4}\right)}^3\cdot x^3=\frac{1}{64}{x}^3$

c)

${\left(\frac{a}{2b^2}\right)}^2$ $=\frac{a^2}{2^2\cdot {\left(b^2\right)}^2}$  = $\frac{a^2}{4b^4}$

${\left(\frac{-2z^3}{0,1x}\right)}^3$ $=\frac{{\left(-2\right)}^3\cdot {\left(z^3\right)}^3}{{\left(0,1\right)}^3\cdot x^3}$ $=$ $\frac{-8z^9}{0,001x^3}$

${\left(\frac{x^2{y}^3{z}^4}{2a^5}\right)}^3$ = $\frac{{\left(x^2\right)}^3\cdot {\left(y^3\right)}^3\cdot {\left(z^4\right)}^3}{2^3\cdot {\left(a^5\right)}^3}$ $=\frac{x^6{y}^9{z}^{12}}{8a^{15}}$