14) \( \frac{3 x^{6} y^{8} z^{5}}{2 x^{8} y^{5} z^{7} \cdot-8 z x^{-4} \cdot-4 x y^{0} z^{0}} \)

To simplify the expression: \[ \frac{3 x^{6} y^{8} z^{5}}{2 x^{8} y^{5} z^{7} \cdot -8 z x^{-4} \cdot -4 x y^{0} z^{0}} \] **Step 1: Simplify the Denominator** First, simplify each part of the denominator: 1. \( y^{0} = 1 \) and \( z^{0} = 1 \), so the third term becomes: \[ -4x \cdot 1 \cdot 1 = -4x \] 2. Multiply the constants and like variables in the denominator: \[ 2 x^{8} y^{5} z^{7} \cdot (-8 z x^{-4}) \cdot (-4x) \] - **Constants**: \( 2 \times (-8) \times (-4) = 64 \) - **Variables**: \[ x^{8} \cdot x^{-4} \cdot x = x^{8-4+1} = x^{5} \] \[ y^{5} \cdot 1 = y^{5} \] \[ z^{7} \cdot z \cdot 1 = z^{8} \] So, the denominator simplifies to: \[ 64 x^{5} y^{5} z^{8} \] **Step 2: Simplify the Entire Fraction** Now, write the entire expression with the simplified denominator: \[ \frac{3 x^{6} y^{8} z^{5}}{64 x^{5} y^{5} z^{8}} \] Subtract the exponents for each variable: - **For \( x \)**: \[ x^{6-5} = x^{1} = x \] - **For \( y \)**: \[ y^{8-5} = y^{3} \] - **For \( z \)**: \[ z^{5-8} = z^{-3} = \frac{1}{z^{3}} \] **Final Simplified Form:** \[ \frac{3x y^{3}}{64 z^{3}} \] **Answer:** \[ \frac{3\, x\, y^{3}}{64\, z^{3}} \]