Question 2 Simplify, the following, giving your answer with positive exponents. (Assume all variables are \( \neq 0 \).) \( \frac{\left(2 a^{3} b-2\right)^{3}}{2 b^{-2}} \div \frac{4 a^{6} b^{-3}}{2^{4} b^{2}} \)

To simplify the expression \[ \frac{\left(2 a^{3} b-2\right)^{3}}{2 b^{-2}} \div \frac{4 a^{6} b^{-3}}{2^{4} b^{2}}, \] we will follow these steps: 1. **Simplify the division**: Dividing by a fraction is the same as multiplying by its reciprocal. Thus, we rewrite the expression as: \[ \frac{\left(2 a^{3} b-2\right)^{3}}{2 b^{-2}} \cdot \frac{2^{4} b^{2}}{4 a^{6} b^{-3}}. \] 2. **Simplify the second fraction**: The second fraction can be simplified: \[ \frac{2^{4} b^{2}}{4 a^{6} b^{-3}} = \frac{16 b^{2}}{4 a^{6} b^{-3}} = \frac{16}{4 a^{6}} \cdot \frac{b^{2}}{b^{-3}} = 4 b^{2 + 3} a^{-6} = 4 b^{5} a^{-6}. \] 3. **Combine the fractions**: Now we can combine the two fractions: \[ \frac{\left(2 a^{3} b-2\right)^{3} \cdot 4 b^{5} a^{-6}}{2 b^{-2}}. \] 4. **Simplify the denominator**: The denominator can be simplified: \[ 2 b^{-2} = 2 \cdot \frac{1}{b^{2}} = \frac{2}{b^{2}}. \] 5. **Combine the fractions**: Now we can rewrite the entire expression: \[ \frac{\left(2 a^{3} b-2\right)^{3} \cdot 4 b^{5} a^{-6}}{\frac{2}{b^{2}}} = \left(2 a^{3} b-2\right)^{3} \cdot 4 b^{5} a^{-6} \cdot \frac{b^{2}}{2}. \] 6. **Simplify further**: The \(2\) in the numerator and denominator cancels out: \[ = \left(2 a^{3} b-2\right)^{3} \cdot 2 b^{5} a^{-6} \cdot b^{2} = \left(2 a^{3} b-2\right)^{3} \cdot 2 b^{7} a^{-6}. \] 7. **Final expression**: The final simplified expression is: \[ 2 b^{7} a^{-6} \left(2 a^{3} b - 2\right)^{3}. \] 8. **Convert to positive exponents**: To express \(a^{-6}\) with a positive exponent, we rewrite it as: \[ \frac{2 b^{7} \left(2 a^{3} b - 2\right)^{3}}{a^{6}}. \] Thus, the final answer is: \[ \frac{2 b^{7} \left(2 a^{3} b - 2\right)^{3}}{a^{6}}. \]