b. \( \frac{3 x}{5 x-15} \cdot \frac{x-3}{9 x^{2}} \)

To simplify the expression: \[ \frac{3x}{5x - 15} \cdot \frac{x - 3}{9x^{2}} \] **Step 1: Factor Where Possible** First, factor the denominators and numerators to identify common factors that can be canceled. 1. **Factor the denominator \(5x - 15\):** \[ 5x - 15 = 5(x - 3) \] 2. **Rewrite the original expression with the factored form:** \[ \frac{3x}{5(x - 3)} \cdot \frac{x - 3}{9x^{2}} \] **Step 2: Multiply the Fractions** Multiply the numerators together and the denominators together: \[ \frac{3x \cdot (x - 3)}{5(x - 3) \cdot 9x^{2}} = \frac{3x(x - 3)}{45x^{2}(x - 3)} \] **Step 3: Simplify by Cancelling Common Factors** 1. **Cancel the common factor \((x - 3)\) from the numerator and denominator:** \[ \frac{3x \cdot \cancel{(x - 3)}}{45x^{2} \cdot \cancel{(x - 3)}} = \frac{3x}{45x^{2}} \] 2. **Simplify the constants and the \(x\) terms:** \[ \frac{3x}{45x^{2}} = \frac{3}{45x} = \frac{1}{15x} \] (Here, \(3/45\) simplifies to \(1/15\), and \(x/x^{2}\) simplifies to \(1/x\)). **Final Simplified Expression:** \[ \frac{1}{15x} \]