5. \( \frac{\left(-3 x^{6} y^{3} z\right)^{2}\left(-15 x^{7} y^{9} z^{5}\right)^{0}}{18 x^{8} y^{9} z^{-4}} \)

To simplify the expression \( \frac{\left(-3 x^{6} y^{3} z\right)^{2}\left(-15 x^{7} y^{9} z^{5}\right)^{0}}{18 x^{8} y^{9} z^{-4}} \), let's break it down step by step. 1. Start with the individual components: - \( \left(-3 x^{6} y^{3} z\right)^{2} = (-3)^{2} (x^{6})^{2} (y^{3})^{2} (z)^{2} = 9 x^{12} y^{6} z^{2} \). - \( \left(-15 x^{7} y^{9} z^{5}\right)^{0} = 1 \) (any non-zero number to the power of 0 is 1). 2. Now substitute these into the original expression: \[ \frac{9 x^{12} y^{6} z^{2} \cdot 1}{18 x^{8} y^{9} z^{-4}} = \frac{9 x^{12} y^{6} z^{2}}{18 x^{8} y^{9} z^{-4}}. \] 3. Simplify the fractions: - The coefficients: \( \frac{9}{18} = \frac{1}{2} \). - The powers of \( x \): \( x^{12}/x^{8} = x^{12-8} = x^{4} \). - The powers of \( y \): \( y^{6}/y^{9} = y^{6-9} = y^{-3} \) (which is equal to \( \frac{1}{y^{3}} \)). - The powers of \( z \): \( z^{2}/z^{-4} = z^{2 - (-4)} = z^{2 + 4} = z^{6} \). Combining all these, we have: \[ \frac{1}{2} x^{4} \cdot \frac{1}{y^{3}} \cdot z^{6} = \frac{x^{4} z^{6}}{2 y^{3}}. \] Thus, the final simplified expression is: \[ \frac{x^{4} z^{6}}{2 y^{3}}. \]