1. **State the problem:** Simplify the expression $$\left( \frac{7}{5^2} + \log_7 \left( \frac{686}{2} \right) \right) \times 3 \sqrt{\frac{125}{8}} \times \frac{5}{11}$$. 2. **Evaluate each part step-by-step:** - Calculate $5^2$: $$5^2 = 25$$ - Simplify the fraction inside the parentheses: $$\frac{7}{25}$$ - Simplify the argument of the logarithm: $$\frac{686}{2} = 343$$ - Recognize that $343 = 7^3$, so: $$\log_7(343) = \log_7(7^3) = 3$$ - Now sum inside the parentheses: $$\frac{7}{25} + 3 = \frac{7}{25} + \frac{75}{25} = \frac{82}{25}$$ 3. **Evaluate the square root term:** $$\sqrt{\frac{125}{8}} = \sqrt{\frac{125}{8}} = \frac{\sqrt{125}}{\sqrt{8}}$$ - Simplify radicals: $$\sqrt{125} = \sqrt{25 \times 5} = 5\sqrt{5}$$ $$\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}$$ - So: $$\frac{5\sqrt{5}}{2\sqrt{2}}$$ 4. **Multiply all parts:** $$\left( \frac{82}{25} \right) \times 3 \times \frac{5\sqrt{5}}{2\sqrt{2}} \times \frac{5}{11}$$ - Combine constants: $$= \frac{82}{25} \times 3 \times \frac{5}{11} \times \frac{5\sqrt{5}}{2\sqrt{2}}$$ - Multiply numerators and denominators: $$= \frac{82 \times 3 \times 5 \times 5 \sqrt{5}}{25 \times 11 \times 2 \sqrt{2}}$$ - Calculate numerator and denominator separately: $$= \frac{82 \times 3 \times 25 \sqrt{5}}{25 \times 11 \times 2 \sqrt{2}}$$ - Cancel $25$ in numerator and denominator: $$= \frac{82 \times 3 \times \cancel{25} \sqrt{5}}{\cancel{25} \times 11 \times 2 \sqrt{2}} = \frac{82 \times 3 \sqrt{5}}{11 \times 2 \sqrt{2}}$$ - Multiply constants: $$= \frac{246 \sqrt{5}}{22 \sqrt{2}}$$ - Simplify fraction: $$= \frac{\cancel{246}^{123} \sqrt{5}}{\cancel{22}^{11} \sqrt{2}} = \frac{123 \sqrt{5}}{11 \sqrt{2}}$$ 5. **Rationalize the denominator:** $$\frac{123 \sqrt{5}}{11 \sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{123 \sqrt{10}}{11 \times 2} = \frac{123 \sqrt{10}}{22}$$ **Final answer:** $$\boxed{\frac{123 \sqrt{10}}{22}}$$