To simplify the given expression [tex]\(\frac{a^3 b^{-2}}{a b^{-4}}\)[/tex]:
1. Rewrite the negative exponents as positive exponents:
Recall that [tex]\( b^{-n} \)[/tex] can be written as [tex]\(\frac{1}{b^n}\)[/tex].
- [tex]\(b^{-2} = \frac{1}{b^2}\)[/tex]
- [tex]\(b^{-4} = \frac{1}{b^4}\)[/tex]
So, [tex]\(\frac{a^3 b^{-2}}{a b^{-4}}\)[/tex] becomes:
[tex]\[
\frac{a^3 \cdot \frac{1}{b^2}}{a \cdot \frac{1}{b^4}} = \frac{a^3 \cdot \frac{1}{b^2}}{a \cdot \frac{1}{b^4}}
\][/tex]
2. Simplify the fractions inside the expression:
Multiply the numerator and the denominator by [tex]\(b^4\)[/tex] to eliminate the fraction in the denominator:
[tex]\[
\frac{a^3 \cdot \frac{1}{b^2}}{a \cdot \frac{1}{b^4}} = \frac{a^3 \cdot b^4 \cdot \frac{1}{b^2}}{a \cdot b^4 \cdot \frac{1}{b^4}} = \frac{a^3 \cdot b^4}{a \cdot b^2}
\][/tex]
3. Combine and cancel common terms:
Simplify the expression:
[tex]\[
\frac{a^3 \cdot b^4}{a \cdot b^2}
\][/tex]
- [tex]\(a^3\)[/tex] divided by [tex]\(a\)[/tex] is [tex]\(a^{3-1} = a^2\)[/tex]
- [tex]\(b^4\)[/tex] divided by [tex]\(b^2\)[/tex] is [tex]\(b^{4-2} = b^2\)[/tex]
So the simplified expression is:
[tex]\[
a^2 b^2
\][/tex]
Therefore, the correct final expression after eliminating the negative exponents and simplifying is:
[tex]\[
\frac{a^3 b^4}{a b^2}
\][/tex]
The corresponding answer to the question is:
[tex]\[
\boxed{\frac{a^3 b^4}{a b^2}}
\][/tex]
