To solve the given expression [tex]\(\frac{4 a^2 b^{-2}}{16 a^{-3} b}\)[/tex] and eliminate the negative exponents, let's go through a step-by-step solution.
Step 1: Rewrite the negative exponents as positive exponents by using the property [tex]\(x^{-n} = \frac{1}{x^n}\)[/tex].
Given expression:
[tex]\[
\frac{4 a^2 b^{-2}}{16 a^{-3} b}
\][/tex]
Rewrite [tex]\(b^{-2}\)[/tex] and [tex]\(a^{-3}\)[/tex] as positive exponents:
[tex]\[
= \frac{4 a^2 \cdot \frac{1}{b^2}}{16 \cdot \frac{1}{a^3} \cdot b}
\][/tex]
Step 2: Simplify the expression by multiplying both the numerator and the denominator by [tex]\(b^2\)[/tex] (to eliminate [tex]\(b^{-2}\)[/tex]) and by [tex]\(a^3\)[/tex] (to eliminate [tex]\(a^{-3}\)[/tex]):
Numerator:
[tex]\[
4 a^2 \cdot \frac{1}{b^2} = \frac{4 a^2}{b^2}
\][/tex]
Denominator:
[tex]\[
16 \cdot \frac{1}{a^3} \cdot b = \frac{16 b}{a^3}
\][/tex]
Step 3: Combine the fractions into a single fraction:
[tex]\[
= \frac{\frac{4 a^2}{b^2}}{\frac{16 b}{a^3}} = \frac{4 a^2}{b^2} \cdot \frac{a^3}{16 b}
\][/tex]
Step 4: Multiply the fractions:
[tex]\[
= \frac{4 a^2 \cdot a^3}{b^2 \cdot 16 b}
\][/tex]
Step 5: Simplify the expressions by performing the multiplications:
[tex]\[
= \frac{4 a^{2+3}}{16 b^{2+1}} = \frac{4 a^5}{16 b^3}
\][/tex]
Step 6: Simplify the constant coefficient [tex]\(\frac{4}{16}\)[/tex]:
[tex]\[
= \frac{a^5}{4 b^3}
\][/tex]
This simplification confirms that:
[tex]\[
\frac{4 a^2 b^{-2}}{16 a^{-3} b} = \frac{a^5}{4 b^3}
\][/tex]
When comparing the correct fraction with the given options, we find that the simplified results match with option:
[tex]\[
\boxed{\frac{4\left(a^2\right)\left(a^3\right)}{16(b)\left(b^2\right)}}
\][/tex]
Thus, the correct answer is:
[tex]\[
\frac{4\left(a^2\right)\left(a^3\right)}{16(b)\left(b^2\right)}
\][/tex]
