What is the quotient \( \frac{15 p^{-4} q^{-6}}{-20 p^{-12} q^{-3}} \) in simplified form? Assume \( p \neq 0, q \neq 0 \)

To simplify the expression \( \frac{15 p^{-4} q^{-6}}{-20 p^{-12} q^{-3}} \), we start by separating the coefficients and the variables: 1. **Coefficients**: \( \frac{15}{-20} = -\frac{3}{4} \) 2. **Variables**: For \( p \) and \( q \): - For \( p \): \( \frac{p^{-4}}{p^{-12}} = p^{-4 - (-12)} = p^{-4 + 12} = p^{8} \) - For \( q \): \( \frac{q^{-6}}{q^{-3}} = q^{-6 - (-3)} = q^{-6 + 3} = q^{-3} \) Now, combining everything, we get: \[ -\frac{3}{4} p^{8} q^{-3} \] To express \( q^{-3} \) as a positive exponent, we rewrite it as \( \frac{1}{q^3} \): \[ -\frac{3}{4} \cdot p^{8} \cdot \frac{1}{q^{3}} = -\frac{3 p^{8}}{4 q^{3}} \] Therefore, the quotient in simplified form is: \[ -\frac{3 p^{8}}{4 q^{3}} \]