1. **State the problem:** Evaluate $$\sqrt[3]{\frac{0.119 \times 0.256}{0.068} \times 7}$$ without using a calculator or table, leaving the answer as a simplified fraction. 2. **Rewrite decimals as fractions:** - $0.119 = \frac{119}{1000}$ - $0.256 = \frac{256}{1000}$ - $0.068 = \frac{68}{1000}$ 3. **Substitute fractions into the expression:** $$\sqrt[3]{\frac{\frac{119}{1000} \times \frac{256}{1000}}{\frac{68}{1000}} \times 7}$$ 4. **Simplify inside the cube root:** Multiply numerator inside the fraction: $$\frac{119 \times 256}{1000 \times 1000} = \frac{30464}{1,000,000}$$ Divide by $\frac{68}{1000}$ is the same as multiplying by its reciprocal: $$\frac{30464}{1,000,000} \times \frac{1000}{68} = \frac{30464 \times 1000}{1,000,000 \times 68} = \frac{30,464,000}{68,000,000}$$ Simplify numerator and denominator by dividing numerator and denominator by 1,000,000: $$\frac{30.464}{68}$$ Multiply by 7: $$\frac{30.464}{68} \times 7 = \frac{213.248}{68}$$ 5. **Convert decimals to fractions for exact simplification:** Recall $30.464 = \frac{30464}{1000}$ and $213.248 = 30.464 \times 7 = \frac{30464}{1000} \times 7 = \frac{213248}{10000}$ So the expression inside the cube root is: $$\frac{213248/10000}{68} = \frac{213248}{10000 \times 68} = \frac{213248}{680000}$$ 6. **Simplify the fraction:** Divide numerator and denominator by 16: $$\frac{213248 \div 16}{680000 \div 16} = \frac{13328}{42500}$$ 7. **Final expression:** $$\sqrt[3]{\frac{13328}{42500}}$$ 8. **Check for perfect cubes:** - $13328 = 2^4 \times 11^3$ (since $11^3=1331$, $1331 \times 10 = 13310$ close, but let's factor 13328 exactly) Factor 13328: - Divide by 8: $13328 \div 8 = 1666$ - 8 is $2^3$, so $13328 = 2^3 \times 1666$ - Factor 1666: $1666 \div 11 = 151.45$ no, try 2: $1666 \div 2 = 833$ - So $13328 = 2^4 \times 833$ - Factor 833: $833 \div 7 = 119$ (since $7 \times 119 = 833$) - Factor 119: $119 = 7 \times 17$ So prime factorization: $$13328 = 2^4 \times 7^2 \times 17$$ Similarly factor 42500: - $42500 = 425 \times 100$ - $425 = 25 \times 17$ - $100 = 2^2 \times 5^2$ So: $$42500 = 2^2 \times 5^2 \times 5^2 \times 17 = 2^2 \times 5^4 \times 17$$ 9. **Rewrite fraction inside cube root:** $$\frac{2^4 \times 7^2 \times 17}{2^2 \times 5^4 \times 17} = 2^{4-2} \times 7^2 \times 17^{1-1} \times 5^{-4} = 2^2 \times 7^2 \times 5^{-4}$$ 10. **Simplify:** $$= \frac{2^2 \times 7^2}{5^4} = \frac{4 \times 49}{625} = \frac{196}{625}$$ 11. **Take cube root:** $$\sqrt[3]{\frac{196}{625}} = \frac{\sqrt[3]{196}}{\sqrt[3]{625}}$$ Note: - $625 = 5^4$, so $\sqrt[3]{625} = 5^{4/3} = 5 \times 5^{1/3}$ - $196 = 14^2$, no perfect cube factors So leave as: $$\frac{\sqrt[3]{196}}{5 \times \sqrt[3]{5}} = \frac{\sqrt[3]{196}}{5 \sqrt[3]{5}}$$ 12. **Final answer:** $$\boxed{\frac{\sqrt[3]{196}}{5 \sqrt[3]{5}}}$$ This is the simplest fractional form without decimals or calculator use.