Let's consider the given expression:
[tex]\[ \frac{b^{-2}}{a b^{-3}} \][/tex]
We need to simplify this expression step-by-step.
1. Rewrite the expression using properties of exponents:
Recall that [tex]\( b^{-n} = \frac{1}{b^n} \)[/tex]. Therefore:
[tex]\[ b^{-2} = \frac{1}{b^2} \quad \text{and} \quad b^{-3} = \frac{1}{b^3} \][/tex]
2. Substitute these into the given expression:
[tex]\[ \frac{b^{-2}}{a b^{-3}} = \frac{\frac{1}{b^2}}{a \cdot \frac{1}{b^3}} \][/tex]
3. Combine the terms in the denominator:
The denominator [tex]\( a \cdot \frac{1}{b^3} \)[/tex] can be written as [tex]\( \frac{a}{b^3} \)[/tex].
4. Rewrite the expression with this substitution:
[tex]\[ \frac{\frac{1}{b^2}}{\frac{a}{b^3}} \][/tex]
5. Simplify the fraction:
Dividing by a fraction is the same as multiplying by its reciprocal. Thus:
[tex]\[ \frac{\frac{1}{b^2}}{\frac{a}{b^3}} = \frac{1}{b^2} \times \frac{b^3}{a} = \frac{1 \cdot b^3}{b^2 \cdot a} = \frac{b^3}{b^2 a} \][/tex]
6. Simplify the expression further:
[tex]\[ \frac{b^3}{b^2 a} = \frac{b^{3-2}}{a} = \frac{b}{a} \][/tex]
Thus, the simplified expression is:
[tex]\[ \boxed{\frac{b}{a}} \][/tex]
