To determine which expression is equivalent to [tex]\(\left(\frac{\left(2 a^{-3} b^4\right)^2}{\left(3 a^5 b\right)^{-2}}\right)^{-1}\)[/tex], let's simplify the expression step by step.
First, simplify the numerator and the denominator separately:
Numerator:
[tex]\[
(2 a^{-3} b^4)^2
\][/tex]
Using the properties of exponents, we can expand this:
[tex]\[
2^2 \cdot (a^{-3})^2 \cdot (b^4)^2 = 4 \cdot a^{-6} \cdot b^8
\][/tex]
So, the simplified form of the numerator is:
[tex]\[
4 a^{-6} b^8
\][/tex]
Denominator:
[tex]\[
(3 a^5 b)^{-2}
\][/tex]
Again, using the properties of exponents:
[tex]\[
(3 a^5 b)^{-2} = 3^{-2} \cdot (a^5)^{-2} \cdot (b)^{-2} = \frac{1}{3^2} \cdot a^{-10} \cdot b^{-2} = \frac{1}{9} a^{-10} b^{-2}
\][/tex]
Next, we substitute the simplified forms of the numerator and denominator back into the original expression:
[tex]\[
\frac{4 a^{-6} b^8}{\frac{1}{9} a^{-10} b^{-2}}
\][/tex]
Simplifying further by multiplying by the reciprocal of the denominator:
[tex]\[
4 a^{-6} b^8 \times 9 a^{10} b^2 = 4 \cdot 9 \cdot a^{-6+10} \cdot b^{8+2} = 36 a^4 b^{10}
\][/tex]
Now, we need to take the reciprocal of this result because of the exponent [tex]\(-1\)[/tex]:
[tex]\[
\left(36 a^4 b^{10}\right)^{-1} = \frac{1}{36 a^4 b^{10}}
\][/tex]
Thus, the given expression simplifies to:
[tex]\[
\frac{1}{36 a^4 b^{10}}
\][/tex]
Comparing this result with the given options, the answer is:
[tex]\[
\boxed{\frac{1}{36 a^4 b^{10}}}
\][/tex]
This corresponds to the third option.
