1. **State the problem:** We need to find the cube root of the expression $$\sqrt[3]{\frac{3.416 \times 0.0789}{\frac{691.6 \times 1.41}{1000}}}$$ using logarithm tables. 2. **Rewrite the expression:** Simplify the denominator first: $$\frac{691.6 \times 1.41}{1000} = \frac{975.156}{1000} = 0.975156$$ So the expression inside the cube root becomes: $$\frac{3.416 \times 0.0789}{0.975156}$$ 3. **Calculate the numerator:** $$3.416 \times 0.0789 = 0.2694324$$ 4. **Calculate the fraction inside the cube root:** $$\frac{0.2694324}{0.975156} \approx 0.2763$$ 5. **Use logarithm tables:** - Find $\log_{10}(0.2763)$. - From tables, $\log_{10}(0.2763) \approx -0.5586$ (since $\log(0.2763) = \log(2.763) - 1$). 6. **Apply the cube root:** The cube root corresponds to raising to the power $\frac{1}{3}$, so: $$\log_{10}(x) = \frac{1}{3} \times (-0.5586) = -0.1862$$ 7. **Find the antilog:** From tables, $\text{antilog}(-0.1862) = 10^{-0.1862} = 0.6515$ approximately. 8. **Final answer:** $$\sqrt[3]{\frac{3.416 \times 0.0789}{\frac{691.6 \times 1.41}{1000}}} \approx 0.6515$$ This means the cube root of the given expression is approximately 0.6515.