1. \( \left[-\left(-\frac{2}{3} a x^{2} y\right):\left(\frac{1}{2} x^{2} y\right)-\frac{1}{2} a^{7}:\left(-\frac{1}{4} a^{6}\right)\right]:\left(-\frac{3}{2} a-\frac{5}{2} a\right) \)
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Let's begin simplifying the expression step by step. Starting with the inside parts of the expression: 1. Simplify \( -\left(-\frac{2}{3} a x^{2} y\right):\left(\frac{1}{2} x^{2} y\right) \). Dividing the two expressions gives: \[ \frac{-\left(-\frac{2}{3} a x^{2} y\right)}{\frac{1}{2} x^{2} y} = \frac{\frac{2}{3} a x^{2} y}{\frac{1}{2} x^{2} y} = \frac{2}{3} a \cdot \frac{2}{1} = \frac{4}{3} a. \] 2. Simplifying \( -\frac{1}{2} a^{7}:\left(-\frac{1}{4} a^{6}\right) \): Dividing yields: \[ \frac{-\frac{1}{2} a^{7}}{-\frac{1}{4} a^{6}} = \frac{\frac{1}{2} a^{7}}{\frac{1}{4} a^{6}} = \frac{1}{2} \cdot \frac{4}{1} \cdot a = 2a. \] 3. Now combine the results from steps 1 and 2: \[ \frac{4}{3} a - 2a. \] To perform this operation, convert \( 2a \) to a fraction with a denominator of 3: \[ 2a = \frac{6}{3} a. \] Thus, \[ \frac{4}{3} a - \frac{6}{3} a = -\frac{2}{3} a. \] 4. Now consider the outer part of the expression, which involves the last part \( -\frac{3}{2} a - \frac{5}{2} a \): \[ -\frac{3}{2} a - \frac{5}{2} a = -\frac{8}{2} a = -4a. \] 5. Now substitute back into the overall expression: \[ \left[-\frac{2}{3} a\right] : \left(-4a\right) = \frac{-\frac{2}{3} a}{-4a} = \frac{\frac{2}{3}}{4} = \frac{2}{3} \cdot \frac{1}{4} = \frac{2}{12} = \frac{1}{6}. \] So, the simplified result of the expression is: \[ \frac{1}{6}. \]
