$\text{a)}$

${\left[{\left(-1\right)}^2\right]}^{31}=1^{31}=1$

${\left(1^3\right)}^{15}=1^{15}=1$

${\left[-{\left(-1\right)}^7\right]}^4={\left[-\left(-1\right)\right]}^4=1^4=1$

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$\text{b)}$

${\left[{\left(-0,1\right)}^2\right]}^3={\left(0,01\right)}^3=0,000001$

${\left({0,2}^3\right)}^2={\left(0,008\right)}^2=0,000064$

${\left[-{\left(-0,1\right)}^2\right]}^3={\left[-0,01\right]}^3=-0,000001$  

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$\text{c)}$

${\left(-3a^{2}b\right)}^3={\left(-3\right)}^3{a}^{2\cdot 3}{b}^3=-27a^6{b}^3$

${\left(-a{b}^{2}c{d}^2\right)}^5={\left(-1\right)}^5{a}^5{b}^{2\cdot 5}{c}^5{d}^{2\cdot 5}$ = $-a^5{b}^{10}{c}^5{d}^{10}$

${\left[{\left(\frac{1}{2}ab\right)}^3\cdot {\left(2a{b}^2\right)}^2\right]}^3$ $={\left(\frac{1}{8}{a}^3{b}^3\cdot 4a^2{b}^4\right)}^3$   $={\left(\frac{1}{2}{a}^5{b}^7\right)}^3=\frac{1}{8}{a}^{15}{b}^{21}$