1. **State the problem:** Simplify the expression $$\frac{\sqrt[5]{\frac{1}{10000}} \times \sqrt[5]{-0.00032}}{\sqrt[4]{(-4)^4}}.$$ 2. **Recall the rules:** - The fifth root of a number $a$ is $a^{\frac{1}{5}}$. - The fourth root of a number $b$ is $b^{\frac{1}{4}}$. - For powers, $(a^m)^n = a^{mn}$. - Negative bases with odd roots are allowed, but even roots of negative numbers are not real. 3. **Simplify each part:** - $\sqrt[5]{\frac{1}{10000}} = \left(10^{-4}\right)^{\frac{1}{5}} = 10^{-\frac{4}{5}}.$ - $\sqrt[5]{-0.00032} = \sqrt[5]{-\frac{32}{100000}} = \sqrt[5]{-\frac{32}{10^5}} = -\sqrt[5]{\frac{32}{10^5}}.$ 4. **Calculate $\sqrt[5]{\frac{32}{10^5}}$:** - $32 = 2^5$, so $\sqrt[5]{32} = 2$. - $\sqrt[5]{10^5} = 10$. - Thus, $\sqrt[5]{\frac{32}{10^5}} = \frac{2}{10} = 0.2$. - So, $\sqrt[5]{-0.00032} = -0.2$. 5. **Calculate numerator:** - $10^{-\frac{4}{5}} \times (-0.2) = -0.2 \times 10^{-\frac{4}{5}}$. 6. **Simplify denominator:** - $(-4)^4 = 4^4 = 256$ (since even power makes it positive). - $\sqrt[4]{256} = 256^{\frac{1}{4}} = (4^4)^{\frac{1}{4}} = 4$. 7. **Combine numerator and denominator:** - $$\frac{-0.2 \times 10^{-\frac{4}{5}}}{4} = -0.05 \times 10^{-\frac{4}{5}}.$$ 8. **Calculate $10^{-\frac{4}{5}}$:** - $10^{-\frac{4}{5}} = 10^{-0.8} = \frac{1}{10^{0.8}}$. - $10^{0.8} \approx 6.3096$. - So, $10^{-0.8} \approx 0.1585$. 9. **Final value:** - $-0.05 \times 0.1585 = -0.007925$. **Answer:** $$\boxed{-0.007925}.$$