b) \( \frac{\left(5 a^{-1} b^{2}\right)^{-2}}{125 a^{5} b^{-3}} \)

To simplify the expression \( \frac{\left(5 a^{-1} b^{2}\right)^{-2}}{125 a^{5} b^{-3}} \), start by simplifying the numerator: 1. Multiply the exponent in the numerator: \[ (5 a^{-1} b^{2})^{-2} = 5^{-2} (a^{-1})^{-2} (b^{2})^{-2} = \frac{1}{25} a^{2} b^{-4} \] 2. Now, substitute this back into the expression: \[ \frac{\frac{1}{25} a^{2} b^{-4}}{125 a^{5} b^{-3}} \] 3. This is equal to: \[ \frac{1}{25} a^{2} b^{-4} \cdot \frac{1}{125 a^{5} b^{-3}} = \frac{1}{25 \cdot 125} \cdot \frac{a^{2}}{a^{5}} \cdot \frac{b^{-4}}{b^{-3}} \] 4. Calculate \(25 \cdot 125 = 3125\): \[ = \frac{1}{3125} \cdot a^{-3} \cdot b^{-1} \] 5. This simplifies to: \[ = \frac{1}{3125 a^{3} b} \] Thus, the simplified expression is: \[ \frac{1}{3125 a^{3} b} \]