Sure, let's simplify the given expression step-by-step using the laws of exponents.
Given expression:
[tex]\[
\left(\frac{a^{-2} b^2}{a^2 b^{-1}}\right)^{-3}
\][/tex]
1. Simplify the expression inside the parentheses:
Inside the parentheses, we have:
[tex]\[
\frac{a^{-2} b^2}{a^2 b^{-1}}
\][/tex]
2. Apply the quotient rule for exponents:
For any base [tex]\( x \)[/tex], [tex]\(\frac{x^m}{x^n} = x^{m-n}\)[/tex].
So, for [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[
\frac{a^{-2}}{a^2} = a^{-2-2} = a^{-4}
\][/tex]
[tex]\[
\frac{b^2}{b^{-1}} = b^{2-(-1)} = b^{2+1} = b^3
\][/tex]
3. Combine the results:
[tex]\[
\frac{a^{-2} b^2}{a^2 b^{-1}} = a^{-4} b^3
\][/tex]
4. Apply the outer exponent (-3):
[tex]\[
\left(a^{-4} b^3\right)^{-3}
\][/tex]
5. Distribute the exponent to each term inside the parentheses:
[tex]\[
(a^{-4})^{-3} \cdot (b^3)^{-3}
\][/tex]
6. Apply the power rule for exponents: [tex]\( (x^m)^n = x^{m \cdot n} \)[/tex]:
[tex]\[
(a^{-4})^{-3} = a^{-4 \cdot (-3)} = a^{12}
\][/tex]
[tex]\[
(b^3)^{-3} = b^{3 \cdot (-3)} = b^{-9}
\][/tex]
7. Combine the results:
[tex]\[
a^{12} \cdot b^{-9}
\][/tex]
Thus, the simplified expression is:
[tex]\[
a^{12} b^{-9}
\][/tex]
