To find the quotient of the given fraction \(\frac{-4 a^{-1} b^5}{20 a^2 b^4}\), we need to perform the following steps:
### Step 1: Rewrite the terms with positive exponents
The numerator is \( -4 a^{-1} b^5 \). Using the property \( a^{-n} = \frac{1}{a^n} \), the numerator can be rewritten as:
[tex]\[
-4 a^{-1} b^5 = -4 \cdot \frac{1}{a} \cdot b^5 = -\frac{4 b^5}{a}
\][/tex]
The denominator is \( 20 a^2 b^4 \). This is already in a simple form:
[tex]\[
20 a^2 b^4
\][/tex]
Therefore, the fraction becomes:
[tex]\[
\frac{-\frac{4 b^5}{a}}{20 a^2 b^4}
\][/tex]
### Step 2: Simplify the complex fraction
To simplify this complex fraction, multiply both the numerator and the denominator by \( a \):
[tex]\[
\frac{-4 b^5}{a} \div 20 a^2 b^4 = \frac{-4 b^5}{a} \cdot \frac{1}{20 a^2 b^4} = \frac{-4 b^5 \cdot 1}{a \cdot 20 a^2 b^4} = \frac{-4 b^5}{20 a^3 b^4}
\][/tex]
### Step 3: Combine the constants and variables
Now, combine and simplify the constants and the variables:
[tex]\[
\frac{-4 b^5}{20 a^3 b^4} = -\frac{4 b^5}{20 a^3 b^4}
\][/tex]
### Step 4: Simplify the constants
[tex]\[
- \frac{4}{20} = -\frac{1}{5}
\][/tex]
So the fraction simplifies to:
[tex]\[
-\frac{1}{5} \cdot \frac{b^5}{a^3 b^4}
\][/tex]
### Step 5: Simplify the variables
Simplify \(\frac{b^5}{b^4}\):
[tex]\[
\frac{b^5}{b^4} = b^{5-4} = b
\][/tex]
Combining this with the remaining terms, we get:
[tex]\[
-\frac{1}{5} \cdot \frac{b}{a^3} = -\frac{b}{5 a^3}
\][/tex]
So, the simplified quotient of the given fraction is:
[tex]\[
-\frac{b}{5 a^3}
\][/tex]
Therefore, the correct answer is:
[tex]\[
-\frac{b}{5 a^3}
\][/tex]
