1. The problem is to solve the equation $$- \frac{8}{9} h = - \frac{4}{27}$$ for $h$. 2. To isolate $h$, divide both sides of the equation by $- \frac{8}{9}$. 3. Dividing by a fraction is the same as multiplying by its reciprocal, so: $$h = \left(- \frac{4}{27}\right) \div \left(- \frac{8}{9}\right) = \left(- \frac{4}{27}\right) \times \left(- \frac{9}{8}\right)$$ 4. Multiply the numerators and denominators: $$h = \frac{-4 \times -9}{27 \times 8} = \frac{36}{216}$$ 5. Simplify the fraction by dividing numerator and denominator by their greatest common divisor, 36: $$h = \frac{\cancel{36}^1}{\cancel{216}^6} = \frac{1}{6}$$ 6. Therefore, the solution is: $$\boxed{h = \frac{1}{6}}$$