To simplify the given expression [tex]\(\frac{a^3 b^{-2}}{a b^{-4}}\)[/tex] and eliminate the negative exponents, let's go step by step.
1. Start with the Original Expression:
[tex]\[
\frac{a^3 b^{-2}}{a b^{-4}}
\][/tex]
2. Simplify the [tex]\(a\)[/tex] terms:
Using the property of exponents [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex], we can simplify the [tex]\(a\)[/tex] terms:
[tex]\[
\frac{a^3}{a} = a^{3-1} = a^2
\][/tex]
So the expression becomes:
[tex]\[
\frac{a^3 b^{-2}}{a b^{-4}} = \frac{a^2 b^{-2}}{b^{-4}}
\][/tex]
3. Simplify the [tex]\(b\)[/tex] terms:
Using the property of exponents [tex]\(\frac{b^m}{b^n} = b^{m-n}\)[/tex], we can simplify the [tex]\(b\)[/tex] terms:
[tex]\[
\frac{b^{-2}}{b^{-4}} = b^{-2 - (-4)} = b^{-2 + 4} = b^2
\][/tex]
So the expression now becomes:
[tex]\[
a^2 b^2
\][/tex]
4. Final Simplified Expression:
Therefore, the expression with eliminated negative exponents is:
[tex]\[
a^2 b^2
\][/tex]
The correct answer from the given options is:
[tex]\[
\frac{a^3 b^{-2}}{a b^{-4}} = a^2 b^2
\][/tex]
Comparing this with the options provided:
- [tex]\(\frac{a^3 b^4}{a b^2}\)[/tex]
Clearly, the correct simplified form from the options given is:
[tex]\[
\boxed{\frac{a^3 b^4}{a b^2}}
\][/tex]
