To find an equivalent expression for [tex]\(\left(\frac{\left(2 a^{-3} b^4\right)^2}{\left(3 a^5 b\right)^{-2}}\right)^{-1}\)[/tex], let's proceed step-by-step to simplify the given expression.
1. Expand the numerator and denominator inside the fraction:
We start with the given expression:
[tex]\[
\left(\frac{\left(2 a^{-3} b^4\right)^2}{\left(3 a^5 b\right)^{-2}}\right)^{-1}
\][/tex]
2. Simplify the numerator [tex]\((2 a^{-3} b^4)^2\)[/tex]:
[tex]\[
(2 a^{-3} b^4)^2 = 2^2 \cdot (a^{-3})^2 \cdot (b^4)^2
\][/tex]
[tex]\[
= 4 \cdot a^{-6} \cdot b^8
\][/tex]
3. Simplify the denominator [tex]\((3 a^5 b)^{-2}\)[/tex]:
[tex]\[
(3 a^5 b)^{-2} = 3^{-2} \cdot (a^5)^{-2} \cdot (b)^{-2}
\][/tex]
[tex]\[
= \frac{1}{3^2} \cdot a^{-10} \cdot b^{-2}
\][/tex]
[tex]\[
= \frac{1}{9} \cdot a^{-10} \cdot b^{-2}
\][/tex]
[tex]\[
= \frac{a^{-10} b^{-2}}{9}
\][/tex]
4. Combine the simplified forms of the numerator and denominator:
[tex]\[
\frac{4 a^{-6} b^8}{\frac{a^{-10} b^{-2}}{9}} = 4 a^{-6} b^8 \times \frac{9}{a^{-10} b^{-2}}
\][/tex]
5. Simplify the multiplication:
[tex]\[
= 4 \times 9 \times a^{-6} \times a^{10} \times b^8 \times b^2
\][/tex]
[tex]\[
= 36 \times a^{-6 + 10} \times b^{8 + 2}
\][/tex]
[tex]\[
= 36 \times a^4 \times b^{10}
\][/tex]
6. Take the inverse (as indicated by the outer exponent [tex]\(-1\)[/tex]):
[tex]\[
\left(36 a^4 b^{10}\right)^{-1} = \frac{1}{36 a^4 b^{10}}
\][/tex]
Therefore, the expression [tex]\(\left(\frac{\left(2 a^{-3} b^4\right)^2}{\left(3 a^5 b\right)^{-2}}\right)^{-1}\)[/tex] simplifies to:
[tex]\[
\boxed{\frac{1}{36 a^4 b^{10}}}
\][/tex]
