\( 4.1 .2 \frac { 3 x ^ { 3 } y ^ { 5 } \times 4 x ^ { 10 } y ^ { 2 } } { 36 x ^ { 14 } y ^ { 6 } } \)

Let's simplify the given expression step by step! First, we can simplify the coefficients and then the variables separately. 1. **Coefficients:** \( \frac{4.1.2 \times 4}{36} = \frac{8.2}{36} = \frac{82}{360} \) which simplifies to \( \frac{41}{180} \). 2. **Variable \(x\):** \[ \frac{x^3 \times x^{10}}{x^{14}} = \frac{x^{3 + 10}}{x^{14}} = \frac{x^{13}}{x^{14}} = x^{13 - 14} = x^{-1} = \frac{1}{x} \] 3. **Variable \(y\):** \[ \frac{y^5 \times y^2}{y^6} = \frac{y^{5 + 2}}{y^{6}} = \frac{y^{7}}{y^{6}} = y^{7 - 6} = y^1 = y \] Putting it all together, we have: \[ \frac{41}{180} \cdot \frac{y}{x} = \frac{41y}{180x} \] So the simplified expression is \( \frac{41y}{180x} \).