To simplify the expression [tex]\(\frac{15 p^{-4} q^{-6}}{-20 p^{-12} q^{-3}}\)[/tex], let's break it down step-by-step:
1. Rewrite the Expression:
[tex]\[
\frac{15 p^{-4} q^{-6}}{-20 p^{-12} q^{-3}}
\][/tex]
2. Combine the Constants:
[tex]\[
\frac{15}{-20}
\][/tex]
Simplify [tex]\(\frac{15}{-20}\)[/tex]:
[tex]\[
\frac{15}{-20} = -\frac{3}{4}
\][/tex]
3. Combine the Powers of [tex]\( p \)[/tex]:
[tex]\[
\frac{p^{-4}}{p^{-12}}
\][/tex]
Using the properties of exponents [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex]:
[tex]\[
p^{-4 - (-12)} = p^{-4 + 12} = p^{8}
\][/tex]
4. Combine the Powers of [tex]\( q \)[/tex]:
[tex]\[
\frac{q^{-6}}{q^{-3}}
\][/tex]
Using the properties of exponents [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex]:
[tex]\[
q^{-6 - (-3)} = q^{-6 + 3} = q^{-3}
\][/tex]
5. Combine All Parts:
Substitute back into the expression:
[tex]\[
-\frac{3}{4} \cdot p^8 \cdot q^{-3}
\][/tex]
6. Rewrite with Positive Exponent for [tex]\( q \)[/tex]:
[tex]\[
-\frac{3 p^8}{4 q^3}
\][/tex]
Thus, the simplified form of the expression [tex]\(\frac{15 p^{-4} q^{-6}}{-20 p^{-12} q^{-3}}\)[/tex] is:
[tex]\[
- \frac{3 p^8}{4 q^3}
\][/tex]
Given the choices, the correct answer is:
[tex]\[
-\frac{3 p^8}{4 q^3}
\][/tex]
