To determine which expression is equivalent to [tex]\(\left( \frac{(2 a^{-3} b^4)^2}{(3 a^5 b)^{-2}} \right)^{-1}\)[/tex], we will proceed step by step:
### Step 1: Simplify the Numerator
First, simplify [tex]\((2 a^{-3} b^4)^2\)[/tex]:
[tex]\[
(2 a^{-3} b^4)^2 = 2^2 \cdot (a^{-3})^2 \cdot (b^4)^2 = 4 \cdot a^{-6} \cdot b^8
\][/tex]
So, the numerator is [tex]\(4 a^{-6} b^8\)[/tex].
### Step 2: Simplify the Denominator
Next, simplify [tex]\((3 a^5 b)^{-2}\)[/tex]:
[tex]\[
(3 a^5 b)^{-2} = \left( \frac{1}{3 a^5 b} \right)^2 = \frac{1}{(3 a^5 b)^2} = \frac{1}{3^2 \cdot (a^5)^2 \cdot (b)^2} = \frac{1}{9 a^{10} b^2}
\][/tex]
So, the denominator is [tex]\(\frac{1}{9 a^{10} b^2}\)[/tex].
### Step 3: Divide the Numerator by the Denominator
Now, divide the simplified numerator by the simplified denominator:
[tex]\[
\frac{4 a^{-6} b^8}{\frac{1}{9 a^{10} b^2}} = 4 a^{-6} b^8 \cdot 9 a^{10} b^2 = 36 a^{-6 + 10} b^{8 + 2}
\][/tex]
Simplify the exponents:
[tex]\[
36 a^{4} b^{10}
\][/tex]
### Step 4: Apply the Inverse
The original expression had an inverse operation indicated by [tex]\(\left( \ldots \right)^{-1}\)[/tex]:
[tex]\[
\left( 36 a^{4} b^{10} \right)^{-1} = \frac{1}{36 a^{4} b^{10}}
\][/tex]
Thus, the expression equivalent to the given expression is:
[tex]\[
\frac{1}{36 a^4 b^{10}}
\][/tex]
Matching this with the given choices, the correct answer is:
[tex]\[
\boxed{\frac{1}{36 a^4 b^{10}}}
\][/tex]
