1. Identify the value of the product of the exponential power on the \( x \) and \( y \) terms. \( \frac{12 x^{5} z^{8} y^{9}}{2 x^{-1} z^{4} y^{4}} \)

To find the product of the exponential powers on the \(x\) and \(y\) terms, we first simplify the given expression: \[ \frac{12 x^{5} z^{8} y^{9}}{2 x^{-1} z^{4} y^{4}} = \frac{12}{2} \cdot \frac{x^{5}}{x^{-1}} \cdot \frac{z^{8}}{z^{4}} \cdot \frac{y^{9}}{y^{4}} \] This simplifies to: \[ 6 \cdot x^{5 - (-1)} \cdot z^{8 - 4} \cdot y^{9 - 4} = 6 \cdot x^{6} \cdot z^{4} \cdot y^{5} \] Now, the value of the product of the exponential powers for \(x\) and \(y\) is: - For \(x\), the power is \(6\). - For \(y\), the power is \(5\). So, the product of the powers is: \[ 6 \cdot 5 = 30 \] Therefore, the value of the product of the exponential powers is \(30\).