a) $2^2\cdot 5^2={\left(2\cdot 5\right)}^2={10}^2=100$  

b) ${\left(\frac{1}{9}\right)}^3\cdot {\left(2\frac{1}{4}\right)}^3={\left(\frac{1}{9}\cdot 2\frac{1}{4}\right)}^3={\left(\frac{1}{9}\cdot \frac{9}{4}\right)}^3={\left(\frac{1}{{\sout{9}}^1}\cdot \frac{{\sout{9}}^1}{4}\right)}^3={\left(\frac{1}{1}\cdot \frac{1}{4}\right)}^3={\left(\frac{1}{4}\right)}^3=\frac{1}{64}$