6. \( \frac{20 p^{-7} q}{4 p^{-2} q^{9} \cdot-2 p^{5} q^{-3}} \)

To simplify the expression \( \frac{20 p^{-7} q}{4 p^{-2} q^{9} \cdot -2 p^{5} q^{-3}} \), let's first focus on simplifying the denominator: 1. Combine the constants: - \( 4 \times -2 = -8 \). 2. Combine the powers of \( p \): - \( p^{-2} \cdot p^{5} = p^{-2 + 5} = p^{3} \). 3. Combine the powers of \( q \): - \( q^{9} \cdot q^{-3} = q^{9 - 3} = q^{6} \). Thus, the denominator becomes: \[ -8 p^{3} q^{6} \] Now, substituting back into the expression: \[ \frac{20 p^{-7} q}{-8 p^{3} q^{6}} \] Now, simplify the constants: \[ \frac{20}{-8} = -\frac{5}{2} \] Now simplify the powers of \( p \): \[ \frac{p^{-7}}{p^{3}} = p^{-7 - 3} = p^{-10} \] And for the powers of \( q \): \[ \frac{q}{q^{6}} = q^{1 - 6} = q^{-5} \] Putting it all together, we have: \[ -\frac{5}{2} p^{-10} q^{-5} \] This can also be expressed using positive exponents: \[ -\frac{5}{2} \cdot \frac{1}{p^{10} q^{5}} \] So the fully simplified expression is: \[ -\frac{5}{2 p^{10} q^{5}} \]