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

To simplify the expression \( \left(4 a^{-\frac{4}{5}} b^{\frac{1}{10}}\right)^{\frac{5}{2}}\left(a^{\frac{1}{3}} b^{\frac{1}{4}}\right)^{3} \), we can break it down step by step. First, let's simplify each term separately: 1. For \( \left(4 a^{-\frac{4}{5}} b^{\frac{1}{10}}\right)^{\frac{5}{2}} \): - \( 4^{\frac{5}{2}} = 4^{2.5} = 2^5 = 32 \) - \( a^{-\frac{4}{5} \cdot \frac{5}{2}} = a^{-10/2} = a^{-5} \) - \( b^{\frac{1}{10} \cdot \frac{5}{2}} = b^{\frac{5}{20}} = b^{\frac{1}{4}} \) Thus, this part simplifies to: \[ 32 a^{-5} b^{\frac{1}{4}} \] 2. Next, simplify \( \left(a^{\frac{1}{3}} b^{\frac{1}{4}}\right)^{3} \): - \( a^{\frac{1}{3} \cdot 3} = a^{1} = a \) - \( b^{\frac{1}{4} \cdot 3} = b^{\frac{3}{4}} \) So this part simplifies to: \[ a b^{\frac{3}{4}} \] Now, combining both parts, we have: \[ 32 a^{-5} b^{\frac{1}{4}} \cdot a b^{\frac{3}{4}} \] When multiplying the variables: - For \( a \): \( a^{-5} \cdot a^1 = a^{-4} \) - For \( b \): \( b^{\frac{1}{4}} \cdot b^{\frac{3}{4}} = b^{\frac{4}{4}} = b^1 = b \) Now we can put everything together: \[ 32 a^{-4} b \] This expression can be rewritten as: \[ \frac{32b}{a^4} \] So the final simplified expression is: \[ \frac{32b}{a^4} \]