To simplify the expression [tex]\(\frac{b^{-2}}{a b^{-3}}\)[/tex], let's go through the steps:
1. Rewrite the Expression:
The given expression is [tex]\(\frac{b^{-2}}{a b^{-3}}\)[/tex].
2. Separate the Fractions:
We can break it down to make it easier to handle:
[tex]\[
\frac{b^{-2}}{a b^{-3}} = \frac{b^{-2}}{a} \cdot \frac{1}{b^{-3}}
\][/tex]
3. Simplify Each Part:
Next, we handle the fractions individually:
[tex]\[
\frac{b^{-2}}{a} \cdot \frac{1}{b^{-3}}
\][/tex]
4. Apply the Laws of Exponents:
Recall that [tex]\(\frac{1}{b^{-3}} = b^{3}\)[/tex]. So the expression now is:
[tex]\[
\frac{b^{-2}}{a} \cdot b^{3}
\][/tex]
5. Combine the Exponents:
Using the laws of exponents [tex]\(\frac{b^m}{b^n} = b^{m-n}\)[/tex], we get:
[tex]\[
b^{-2 + 3} = b^{1} = b
\][/tex]
6. Rewrite the Expression:
Combining all the parts, we have:
[tex]\[
\frac{b}{a}
\][/tex]
Therefore, the equivalent expression to [tex]\(\frac{b^{-2}}{a b^{-3}}\)[/tex] is:
[tex]\[
\boxed{\frac{b}{a}}
\][/tex]
