To find the quotient \(\frac{15 p^{-4} q^{-6}}{-20 p^{-12} q^{-3}}\) in simplified form, we will simplify both the coefficients and the exponents.
1. Simplify the coefficient:
[tex]\[
\frac{15}{-20} = \frac{15}{-20} = -\frac{3}{4}
\][/tex]
2. Simplify the exponents of \(p\):
[tex]\[
p^{-4} \div p^{-12} = p^{-4 - (-12)} = p^{-4 + 12} = p^{8}
\][/tex]
3. Simplify the exponents of \(q\):
[tex]\[
q^{-6} \div q^{-3} = q^{-6 - (-3)} = q^{-6 + 3} = q^{-3}
\][/tex]
Thus, combining these simplifications, the quotient can be written as:
[tex]\[
-\frac{3 p^8}{4 q^3}
\][/tex]
So, the simplified form of [tex]\(\frac{15 p^{-4} q^{-6}}{-20 p^{-12} q^{-3}}\)[/tex] is [tex]\(\boxed{-\frac{3 p^8}{4 q^3}}\)[/tex].
