13. \( \frac{x^{2}+8 x+7}{12 x^{3}} \div \frac{x+7}{6 x} \)

To solve the expression \( \frac{x^{2}+8 x+7}{12 x^{3}} \div \frac{x+7}{6 x} \), we first rewrite the division as multiplication by the reciprocal: \[ \frac{x^{2} + 8x + 7}{12x^{3}} \cdot \frac{6x}{x+7} \] Next, we notice that \(x^{2} + 8x + 7\) can be factored. We look for two numbers that multiply to 7 and add to 8. These numbers are 1 and 7, so we can factor it as: \[ x^{2} + 8x + 7 = (x + 1)(x + 7) \] Now our expression becomes: \[ \frac{(x + 1)(x + 7)}{12x^{3}} \cdot \frac{6x}{x + 7} \] Now we can cancel out the \(x + 7\) term from the numerator and denominator: \[ \frac{(x + 1)}{12x^{3}} \cdot 6x \] Multiplying across, we have: \[ \frac{(x + 1) \cdot 6x}{12x^{3}} = \frac{6x(x + 1)}{12x^{3}} \] Now, simplifying the fraction, we notice that \(6\) and \(12\) simplify to \(\frac{1}{2}\) and we can also reduce \(x\) from the numerator and the denominator: \[ \frac{x + 1}{2x^2} \] So the final simplified expression is: \[ \frac{x + 1}{2x^2} \]