$\text{a)}\frac{1}{3}{\left[g\left(4\right)-g\left(16\right)\right]}^{g\left(1\right)}=\frac{1}{3}\cdot {\left[-2-\left(-4\right)\right]}^{-1}=\frac{1}{3}\cdot {\left[-2+4\right]}^{-1}=\frac{1}{3}\cdot 2^{-1}=\frac{1}{3}\cdot \frac{1}{2}=\frac{1}{6}$