To solve the given problem, we need to simplify the expression [tex]\(\frac{4 a^2 b^{-2}}{16 a^{-3} b}\)[/tex] and eliminate the negative exponents. Let's go through the steps methodically:
### Step 1: Simplify the Coefficients
First, simplify the numerical coefficients:
[tex]\[
\frac{4}{16} = \frac{1}{4}
\][/tex]
### Step 2: Apply Properties of Exponents
Use the properties of exponents to simplify the variables:
#### For [tex]\(a\)[/tex]:
[tex]\[
a^{2} \text{ in the numerator and } a^{-3} \text{ in the denominator can be combined as: }
\][/tex]
[tex]\[
a^{2 - (-3)} = a^{2 + 3} = a^{5}
\][/tex]
#### For [tex]\(b\)[/tex]:
[tex]\[
b^{-2} \text{ in the numerator and } b \text{ in the denominator can be combined as: }
\][/tex]
[tex]\[
b^{-2 - 1} = b^{-3}
\][/tex]
### Step 3: Combine the Results
[tex]\[
\frac{4 a^2 b^{-2}}{16 a^{-3} b} = \frac{1}{4} a^{5} b^{-3}
\][/tex]
### Step 4: Eliminate the Negative Exponents
The final step is to eliminate the negative exponents. To do this, we use the property that [tex]\(x^{-n} = \frac{1}{x^{n}}\)[/tex].
[tex]\[
\frac{1}{4} a^{5} b^{-3} = \frac{a^{5}}{4 b^{3}}
\][/tex]
We now have the expression with negative exponents eliminated.
[tex]\[
\frac{a^{5}}{4 b^{3}} = \frac{1}{4 a^{-5} b^{3}}
\][/tex]
This matches the option:
[tex]\[
\boxed{\frac{1}{4 a b^{3}}}
\][/tex]
So the correct answer is:
[tex]\[
\boxed{\frac{1}{4 a b^3}}
\][/tex]
