We are given the expression:

$\left( \frac{4}{9abc^3} \right) \times \left( \frac{-27}{5a^3b^2} \right) \times \left( \frac{-8b^3}{c} \right)$

Let's simplify this step by step.

Step 1: Multiply the constants (numerators and denominators)

We multiply the numerical coefficients separately:

$\frac{4}{9} \times \frac{-27}{5} \times \frac{-8}{1}$

\quad \text{and} \quad -108 \times (-8) = 864$$ For the denominators: \[ 9 \times 5 \times