(b) \( \frac{18 a^{3} b}{3 u v^{3}} \times \frac{2 u^{2} v}{9 b^{2}} \div \frac{4 u^{3} v}{12 b v^{2}} \)

To solve the expression \( \frac{18 a^{3} b}{3 u v^{3}} \times \frac{2 u^{2} v}{9 b^{2}} \div \frac{4 u^{3} v}{12 b v^{2}} \), start by simplifying each part step-by-step. 1. **Convert the division to multiplication by the reciprocal**: \[ = \frac{18 a^{3} b}{3 u v^{3}} \times \frac{2 u^{2} v}{9 b^{2}} \times \frac{12 b v^{2}}{4 u^{3} v} \] 2. **Now, multiply the first two fractions**: \[ = \frac{(18 a^{3} b) \cdot (2 u^{2} v)}{(3 u v^{3}) \cdot (9 b^{2})} \] Simplifying the numerator: \[ = \frac{36 a^{3} b u^{2} v}{27 u b^{2} v^{3}} \] We can simplify this further: \[ = \frac{4 a^{3} u^{2}}{3 b v^{2}} \] 3. **Now multiply with the last fraction**: \[ = \frac{4 a^{3} u^{2}}{3 b v^{2}} \times \frac{12 b v^{2}}{4 u^{3} v} \] The numerator and denominator simplify as: \[ = \frac{48 a^{3} u^{2} b v^{2}}{12 u^{3} b v} \] Cancelling \(b\), \(v^{2}\), and simplifying: \[ = \frac{4 a^{3} u^{2}}{u^{3}} = \frac{4 a^{3}}{u} \] Thus, the final simplified result is: \[ \frac{4 a^{3}}{u} \]