20. \( \left[a^{6} b^{2} \cdot\left(-\frac{1}{4} a b^{2}\right)+\frac{3}{2} a^{3} b^{2} \cdot\left(-\frac{1}{3} a^{2} b\right)^{2}\right]:\left[\frac{1}{4} a b \cdot\left(-a^{3} b\right)^{2}\right]+\frac{1}{3} b \)

To simplify the expression, let's break it down step by step. 1. The numerator is: \[ a^{6} b^{2} \cdot \left(-\frac{1}{4} a b^{2}\right) + \frac{3}{2} a^{3} b^{2} \cdot \left(-\frac{1}{3} a^{2} b\right)^{2} \] First, simplify each term: - The first term simplifies to: \[ -\frac{1}{4} a^{7} b^{4} \] - The second term: \[ \left(-\frac{1}{3} a^{2} b\right)^{2} = \frac{1}{9} a^4 b^2 \quad \text{so this term becomes:} \] \[ \frac{3}{2} a^{3} b^{2} \cdot \frac{1}{9} a^4 b^2 = \frac{3}{18} a^{7} b^{4} = \frac{1}{6} a^{7} b^{4} \] Combining the two terms in the numerator: \[ -\frac{1}{4} a^{7} b^{4} + \frac{1}{6} a^{7} b^{4} \] To add these fractions, find a common denominator (12): \[ -\frac{3}{12} a^{7} b^{4} + \frac{2}{12} a^{7} b^{4} = -\frac{1}{12} a^{7} b^{4} \] 2. Now, for the denominator: \[ \frac{1}{4} a b \cdot (-a^{3} b)^{2} = \frac{1}{4} a b \cdot a^{6} b^{2} = \frac{1}{4} a^{7} b^{3} \] Putting it all together, the expression now looks like: \[ \frac{-\frac{1}{12} a^{7} b^{4}}{\frac{1}{4} a^{7} b^{3}} + \frac{1}{3} b \] 3. Simplifying the fraction: \[ = -\frac{1}{12} \cdot \frac{4}{1} b = -\frac{4}{12} b = -\frac{1}{3} b \] 4. Finally, add the remaining term: \[ -\frac{1}{3} b + \frac{1}{3} b = 0 \] So the final result of the expression is: \[ \boxed{0} \]