d) \( \left(4 a^{-\frac{4}{5}} b^{\frac{1}{10}}\right)^{\frac{5}{2}}\left(a^{\frac{1}{3}} b^{\frac{1}{4}}\right)^{3} \)

To solve the expression, let’s break it down step by step. First, we deal with the first part: \[ \left(4 a^{-\frac{4}{5}} b^{\frac{1}{10}}\right)^{\frac{5}{2}} = 4^{\frac{5}{2}} \cdot a^{-\frac{4}{5} \cdot \frac{5}{2}} \cdot b^{\frac{1}{10} \cdot \frac{5}{2}} = 4^{\frac{5}{2}} \cdot a^{-10/2} \cdot b^{5/20} = 4^{\frac{5}{2}} \cdot a^{-2} \cdot b^{\frac{1}{4}} = 32 \cdot a^{-2} \cdot b^{\frac{1}{4}} \] Next, we simplify the second part: \[ \left(a^{\frac{1}{3}} b^{\frac{1}{4}}\right)^{3} = a^{\frac{1}{3} \cdot 3} \cdot b^{\frac{1}{4} \cdot 3} = a^{1} \cdot b^{\frac{3}{4}} = a^{1} \cdot b^{\frac{3}{4}} \] Now we combine both parts: \[ 32 \cdot a^{-2} \cdot b^{\frac{1}{4}} \cdot a^{1} \cdot b^{\frac{3}{4}} = 32 \cdot a^{-2 + 1} \cdot b^{\frac{1}{4} + \frac{3}{4}} = 32 \cdot a^{-1} \cdot b^{1} \] Thus, the final result is: \[ \frac{32b}{a} \]