$\text{a)}{\left(1\frac{1}{4}\right)}^3={\left(\frac{5}{4}\right)}^3=\frac{5}{4}\cdot \frac{5}{4}\cdot \frac{5}{4}=\frac{125}{64}=1\frac{61}{64}$

$\text{b)}{\left(2\frac{2}{3}\right)}^2={\left(\frac{8}{3}\right)}^2=\frac{8}{3}\cdot \frac{8}{3}=\frac{64}{9}=7\frac{1}{9}$

$\text{c)}{\left(-1\frac{1}{2}\right)}^3={\left(-\frac{3}{2}\right)}^3=\left(-\frac{3}{2}\right)\cdot \left(-\frac{3}{2}\right)\cdot \left(-\frac{3}{2}\right)=-\frac{27}{8}=-3\frac{3}{8}$

$\text{d)}{\left(-2\frac{2}{5}\right)}^2={\left(-\frac{12}{5}\right)}^2=\frac{144}{25}=5\frac{19}{25}$