To simplify the expression \( 18x^3y^2z \cdot \left(-\frac{2}{9}xy^2zx^2\right) \cdot (-2xyzxyx) \), we'll multiply everything together step by step, focusing on both numerical coefficients and variables.
Multiply the coefficients: \[ 18 \cdot \left(-\frac{2}{9}\right) \cdot (-2) = 18 \cdot \left(-\frac{2 \cdot -2}{9}\right) = 18 \cdot \frac{4}{9} \]
To simplify \( 18 \cdot \frac{4}{9} \): \[ 18 = 2 \cdot 9 \quad \Rightarrow \quad \frac{18}{9} = 2 \] \[ 2 \cdot 4 = 8 \]
Multiply the variables: \[ x^3 \cdot x \cdot x = x^{3+1+1} = x^5 \] \[ y^2 \cdot y^2 \cdot y \cdot y = y^{2+2+1+1} = y^6 \] \[ z \cdot z \cdot z = z^{1+1+1} = z^3 \]
Combine all together: Now that we have the coefficient and the powers of the variables, we put everything together: \[ 8x^5y^6z^3 \]
Thus, the simplified expression is: \[ \boxed{8x^5y^6z^3} \]