31. \( \left(-\frac{m}{b^{3}}\right)^{6} \)

To simplify \( \left(-\frac{m}{b^{3}}\right)^{6} \), we apply the power of a quotient and power of a product rules. First, raise both the numerator and denominator to the 6th power: \[ \left(-\frac{m}{b^{3}}\right)^{6} = \frac{(-m)^{6}}{(b^{3})^{6}}. \] Calculating each part yields: \[ (-m)^{6} = m^{6} \quad \text{(since even powers of negative numbers are positive)}, \] and \[ (b^{3})^{6} = b^{18} \quad \text{(using the power of a power rule)}. \] Thus, the expression simplifies to: \[ \frac{m^{6}}{b^{18}}. \] So the final result is: \[ \left(-\frac{m}{b^{3}}\right)^{6} = \frac{m^{6}}{b^{18}}. \]