To simplify the expression \( 18x^3y^2z \cdot \left(-\frac{2}{9}xy^2zx^2\right) \cdot (-2xyzxyx) \), we proceed step by step.
Combine the constants:
- Constant terms: \( 18 \cdot \left(-\frac{2}{9}\right) \cdot (-2) \)
First, simplify the constants:
\[ 18 \cdot \left(-\frac{2}{9}\right) = -\frac{36}{9} = -4 \]
Now, multiplying by \(-2\):
\[ -4 \cdot -2 = 8 \]
Combine the variable parts:
Now let’s combine the variable parts:
\[ x^3 \cdot x \cdot x = x^{3+1+1} = x^5 \]
\[ y^2 \cdot y^2 \cdot y \cdot y = y^{2+2+1+1} = y^6 \]
\[ z \cdot z \cdot z = z^{1+1+1} = z^3 \]
Putting it all together:
Now combine the constant and variable parts together:
\[ 8 \cdot x^5 \cdot y^6 \cdot z^3 = 8x^5y^6z^3 \]
Thus, the simplified expression is:
\[ \boxed{8x^5y^6z^3} \]