1. **State the problem:** Simplify the expression $$\left(4a^{-\frac{4}{5}} b^{\frac{1}{10}}\right)^{\frac{5}{2}} \left(a^{\frac{1}{3}} b^{\frac{1}{4}}\right)^3$$. 2. **Recall the power of a product rule:** $$(xy)^n = x^n y^n$$ and the power of a power rule: $$(x^m)^n = x^{mn}$$. 3. Apply the power to each factor inside the first parentheses: $$\left(4a^{-\frac{4}{5}} b^{\frac{1}{10}}\right)^{\frac{5}{2}} = 4^{\frac{5}{2}} \cdot a^{-\frac{4}{5} \cdot \frac{5}{2}} \cdot b^{\frac{1}{10} \cdot \frac{5}{2}}$$ 4. Calculate each exponent: - $4^{\frac{5}{2}} = (2^2)^{\frac{5}{2}} = 2^{2 \cdot \frac{5}{2}} = 2^5 = 32$ - $a^{-\frac{4}{5} \cdot \frac{5}{2}} = a^{-2}$ - $b^{\frac{1}{10} \cdot \frac{5}{2}} = b^{\frac{1}{4}}$ So the first part simplifies to: $$32 a^{-2} b^{\frac{1}{4}}$$ 5. Now simplify the second parentheses: $$\left(a^{\frac{1}{3}} b^{\frac{1}{4}}\right)^3 = a^{\frac{1}{3} \cdot 3} b^{\frac{1}{4} \cdot 3} = a^1 b^{\frac{3}{4}}$$ 6. Multiply the two results: $$32 a^{-2} b^{\frac{1}{4}} \cdot a^1 b^{\frac{3}{4}} = 32 a^{-2+1} b^{\frac{1}{4} + \frac{3}{4}} = 32 a^{-1} b^1$$ 7. Rewrite with positive exponents: $$32 \frac{b}{a}$$ **Final answer:** $$\boxed{32 \frac{b}{a}}$$