1. **State the problem:** Simplify the expression $$\frac{0.1^{0.4} \cdot 0.2^{0.3} \cdot 0.3^{0.2} \cdot 0.4^{0.1}}{0.5^{0.5} \cdot 0.3^{0.8}}$$. 2. **Rewrite the expression:** Group the terms with the same base if possible. 3. **Note the bases and exponents:** The numerator has bases 0.1, 0.2, 0.3, 0.4 with exponents 0.4, 0.3, 0.2, 0.1 respectively. The denominator has bases 0.5 and 0.3 with exponents 0.5 and 0.8 respectively. 4. **Combine powers with the same base:** The base 0.3 appears in numerator and denominator, so combine exponents: $$0.3^{0.2} \div 0.3^{0.8} = 0.3^{0.2 - 0.8} = 0.3^{-0.6}$$ 5. **Rewrite the entire expression:** $$0.1^{0.4} \cdot 0.2^{0.3} \cdot 0.4^{0.1} \cdot 0.3^{-0.6} \cdot 0.5^{-0.5}$$ 6. **Express all bases as powers of 10 if possible:** - $0.1 = 10^{-1}$ - $0.2 = 2 \times 10^{-1}$ - $0.4 = 4 \times 10^{-1}$ - $0.3 = 3 \times 10^{-1}$ - $0.5 = 5 \times 10^{-1}$ 7. **Rewrite each term:** $$0.1^{0.4} = (10^{-1})^{0.4} = 10^{-0.4}$$ $$0.2^{0.3} = (2 \times 10^{-1})^{0.3} = 2^{0.3} \times 10^{-0.3}$$ $$0.4^{0.1} = (4 \times 10^{-1})^{0.1} = 4^{0.1} \times 10^{-0.1}$$ $$0.3^{-0.6} = (3 \times 10^{-1})^{-0.6} = 3^{-0.6} \times 10^{0.6}$$ $$0.5^{-0.5} = (5 \times 10^{-1})^{-0.5} = 5^{-0.5} \times 10^{0.5}$$ 8. **Combine all powers of 10:** $$10^{-0.4} \times 10^{-0.3} \times 10^{-0.1} \times 10^{0.6} \times 10^{0.5} = 10^{-0.4 - 0.3 - 0.1 + 0.6 + 0.5} = 10^{0.3}$$ 9. **Combine the other factors:** $$2^{0.3} \times 4^{0.1} \times 3^{-0.6} \times 5^{-0.5}$$ 10. **Calculate approximate values:** - $2^{0.3} \approx e^{0.3 \ln 2} \approx e^{0.2079} \approx 1.231$ - $4^{0.1} = (2^2)^{0.1} = 2^{0.2} \approx e^{0.2 \ln 2} \approx e^{0.1386} \approx 1.149$ - $3^{-0.6} = \frac{1}{3^{0.6}} \approx \frac{1}{e^{0.6 \ln 3}} \approx \frac{1}{e^{0.659}} \approx 0.517$ - $5^{-0.5} = \frac{1}{\sqrt{5}} \approx 0.447$ 11. **Multiply these values:** $$1.231 \times 1.149 \times 0.517 \times 0.447 \approx 0.327$$ 12. **Multiply by $10^{0.3}$:** $$10^{0.3} = e^{0.3 \ln 10} \approx e^{0.6908} \approx 1.995$$ 13. **Final value:** $$1.995 \times 0.327 \approx 0.652$$ **Answer:** The value of the expression is approximately **0.652**.