Fully simplify using only positive exponents. \[ \frac{30 x^{8} y^{4}}{12 x^{5} y^{6}} \]

To simplify the expression \(\frac{30 x^{8} y^{4}}{12 x^{5} y^{6}}\), we begin by simplifying the numerical coefficients and then the variables. 1. **Numerical Coefficient**: \( \frac{30}{12} = 2.5 \) or \( \frac{5}{2} \). 2. **Variable \(x\)**: Use the power of a quotient rule, \(x^{m}/x^{n} = x^{m-n}\): \[ x^{8}/x^{5} = x^{8-5} = x^{3} \] 3. **Variable \(y\)**: Similarly, \[ y^{4}/y^{6} = y^{4-6} = y^{-2} \] Now, combining these results, we have: \[ \frac{30 x^{8} y^{4}}{12 x^{5} y^{6}} = \frac{5}{2} x^{3} y^{-2} \] Since we only want positive exponents, we rewrite \(y^{-2}\) as \(\frac{1}{y^{2}}\): \[ \frac{5}{2} x^{3} \frac{1}{y^{2}} = \frac{5 x^{3}}{2 y^{2}} \] Thus, the fully simplified expression is: \[ \frac{5 x^{3}}{2 y^{2}} \]