Let's simplify the expression [tex]\(\frac{15 p^{-4} q^{-6}}{-20 p^{-12} q^{-3}}\)[/tex] step-by-step.
1. Coefficients:
- The coefficient in the numerator is [tex]\(15\)[/tex].
- The coefficient in the denominator is [tex]\(-20\)[/tex].
- Simplify the coefficient fraction [tex]\(\frac{15}{-20}\)[/tex]:
[tex]\[
\frac{15}{-20} = -\frac{15}{20} = -\frac{3}{4}.
\][/tex]
2. Powers of [tex]\(p\)[/tex]:
- In the numerator, we have [tex]\(p^{-4}\)[/tex].
- In the denominator, we have [tex]\(p^{-12}\)[/tex].
- When dividing like bases, subtract the exponents (numerator exponent minus denominator exponent):
[tex]\[
p^{-4} \div p^{-12} = p^{-4 - (-12)} = p^{-4 + 12} = p^{8}.
\][/tex]
3. Powers of [tex]\(q\)[/tex]:
- In the numerator, we have [tex]\(q^{-6}\)[/tex].
- In the denominator, we have [tex]\(q^{-3}\)[/tex].
- When dividing like bases, subtract the exponents (numerator exponent minus denominator exponent):
[tex]\[
q^{-6} \div q^{-3} = q^{-6 - (-3)} = q^{-6 + 3} = q^{-3}.
\][/tex]
Putting everything together:
- The simplified coefficient is [tex]\(-\frac{3}{4}\)[/tex].
- The power of [tex]\(p\)[/tex] is [tex]\(8\)[/tex].
- The power of [tex]\(q\)[/tex] is [tex]\(-3\)[/tex].
Thus, the simplified form of the expression is:
[tex]\[
-\frac{3}{4} \cdot p^{8} \cdot q^{-3}.
\][/tex]
Since [tex]\(q^{-3}\)[/tex] can be written as [tex]\(\frac{1}{q^{3}}\)[/tex], the expression becomes:
[tex]\[
-\frac{3 p^{8}}{4 q^{3}}.
\][/tex]
Therefore, the simplified form of the given expression is:
[tex]\[
-\frac{3 p^8}{4 q^3}.
\][/tex]
The correct answer is [tex]\(\boxed{-\frac{3 p^8}{4 q^3}}\)[/tex].
