$\text{a)}{9}^{\frac{2}{3}}\cdot 9^{-\frac{1}{6}}=9^{\frac{2}{3}-\frac{1}{6}}=9^{\frac{4}{6}-\frac{1}{6}}=9^{\frac{3}{6}}=9^{\frac{1}{2}}=\sqrt{9}=3$      

$\text{b)}{8}^2:8^{\frac{1}{3}}=8^{2-\frac{1}{3}}=8^{1\frac{2}{3}}={\left(2^3\right)}^{\frac{5}{3}}=2^{3\cdot \frac{5}{3}}=2^5=32$  

$\text{c)}2:2^{-1}\cdot {625}^{-\frac{3}{4}}\cdot {625}^{-\frac{1}{4}}=2^{1-\left(-1\right)}\cdot {625}^{-\frac{3}{4}-\frac{1}{4}}=2^2\cdot {625}^{-1}=4\cdot \frac{1}{625}=\frac{4}{625}$  

$\text{d)}{\left({11}^2\right)}^{-\frac{1}{2}}\cdot 1^{90}={11}^{2\cdot \left(-\frac{1}{2}\right)}\cdot 1^{90}=2^{-1}\cdot 1^{90}=\frac{1}{2}\cdot 1=\frac{1}{2}$