Answer :
Let's simplify the given expression step-by-step.
The original expression is:
[tex]\[ \frac{15 p^{-4} q^{-6}}{-20 p^{-12} q^{-3}} \][/tex]
### Step 1: Simplify the coefficient
Separate the numerical coefficients from the variables:
[tex]\[ \frac{15}{-20} \][/tex]
The coefficient simplifies to:
[tex]\[ \frac{15}{-20} = -\frac{3}{4} \][/tex]
### Step 2: Simplify the powers of \( p \)
Use the properties of exponents to simplify:
[tex]\[ \frac{p^{-4}}{p^{-12}} = p^{-4 - (-12)} = p^{-4 + 12} = p^{8} \][/tex]
### Step 3: Simplify the powers of \( q \)
Similarly, simplify the exponents of \( q \):
[tex]\[ \frac{q^{-6}}{q^{-3}} = q^{-6 - (-3)} = q^{-6 + 3} = q^{-3} \][/tex]
### Step 4: Combine the simplified parts
Combine the simplified coefficient and the variables:
[tex]\[ -\frac{3}{4} \cdot p^{8} \cdot q^{-3} \][/tex]
### Step 5: Express the final result in a proper fraction
Rewrite \( q^{-3} \) as \( \frac{1}{q^3} \) to express the entire expression as a single fraction:
[tex]\[ -\frac{3 p^8}{4 q^3} \][/tex]
Hence, the simplified form of the expression \(\frac{15 p^{-4} q^{-6}}{-20 p^{-12} q^{-3}}\) is:
[tex]\[ -\frac{3p^8}{4 q^3} \][/tex]
Among the given options, the correct answer is:
\[
-\frac{3p^8}{4 q^3}
\
The original expression is:
[tex]\[ \frac{15 p^{-4} q^{-6}}{-20 p^{-12} q^{-3}} \][/tex]
### Step 1: Simplify the coefficient
Separate the numerical coefficients from the variables:
[tex]\[ \frac{15}{-20} \][/tex]
The coefficient simplifies to:
[tex]\[ \frac{15}{-20} = -\frac{3}{4} \][/tex]
### Step 2: Simplify the powers of \( p \)
Use the properties of exponents to simplify:
[tex]\[ \frac{p^{-4}}{p^{-12}} = p^{-4 - (-12)} = p^{-4 + 12} = p^{8} \][/tex]
### Step 3: Simplify the powers of \( q \)
Similarly, simplify the exponents of \( q \):
[tex]\[ \frac{q^{-6}}{q^{-3}} = q^{-6 - (-3)} = q^{-6 + 3} = q^{-3} \][/tex]
### Step 4: Combine the simplified parts
Combine the simplified coefficient and the variables:
[tex]\[ -\frac{3}{4} \cdot p^{8} \cdot q^{-3} \][/tex]
### Step 5: Express the final result in a proper fraction
Rewrite \( q^{-3} \) as \( \frac{1}{q^3} \) to express the entire expression as a single fraction:
[tex]\[ -\frac{3 p^8}{4 q^3} \][/tex]
Hence, the simplified form of the expression \(\frac{15 p^{-4} q^{-6}}{-20 p^{-12} q^{-3}}\) is:
[tex]\[ -\frac{3p^8}{4 q^3} \][/tex]
Among the given options, the correct answer is:
\[
-\frac{3p^8}{4 q^3}
\
