Let's simplify the given expression step by step:
The given expression is
[tex]\[
\left(\frac{\left(2 a^{-3} b^4\right)^2}{\left(3 a^5 b\right)^{-2}}\right)^{-1}
\][/tex]
### Step 1: Simplify the Numerator
First, we simplify [tex]\((2 a^{-3} b^4)^2\)[/tex]:
[tex]\[
(2 a^{-3} b^4)^2 = 2^2 \cdot (a^{-3})^2 \cdot (b^4)^2
\][/tex]
Applying the powers, we get:
[tex]\[
2^2 = 4, \quad (a^{-3})^2 = a^{-6}, \quad (b^4)^2 = b^8
\][/tex]
Thus, the numerator becomes:
[tex]\[
4 a^{-6} b^8
\][/tex]
### Step 2: Simplify the Denominator
Next, simplify [tex]\((3 a^5 b)^{-2}\)[/tex]:
[tex]\[
(3 a^5 b)^{-2} = 3^{-2} \cdot (a^5)^{-2} \cdot (b)^{-2}
\][/tex]
Applying the powers, we get:
[tex]\[
3^{-2} = \frac{1}{9}, \quad (a^5)^{-2} = a^{-10}, \quad b^{-2}
\][/tex]
Thus, the denominator becomes:
[tex]\[
\frac{1}{9} a^{-10} b^{-2}
\][/tex]
### Step 3: Combine the Simplified Numerator and Denominator
Next, we combine the simplified numerator and denominator:
[tex]\[
\left(\frac{4 a^{-6} b^8}{\frac{1}{9} a^{-10} b^{-2}}\right)
\][/tex]
Simplify the division:
[tex]\[
\frac{4 a^{-6} b^8}{\frac{1}{9} a^{-10} b^{-2}} = \frac{4 a^{-6} b^8 \cdot 9}{a^{-10} b^{-2}} = 4 \cdot 9 \cdot \frac{a^{-6}}{a^{-10}} \cdot \frac{b^8}{b^{-2}}
\][/tex]
This simplifies to:
[tex]\[
36 \cdot a^{-6 - (-10)} \cdot b^{8 - (-2)} = 36 \cdot a^4 \cdot b^{10}
\][/tex]
### Step 4: Apply the Power of -1 to the Simplified Expression
Next, we apply the power of -1:
[tex]\[
\left(36 a^4 b^{10}\right)^{-1} = \frac{1}{36 a^4 b^{10}}
\][/tex]
### Conclusion
The given multiple-choice answers are:
- [tex]\(\frac{2}{3 a^4 b^{10}}\)[/tex]
- [tex]\(\frac{4}{9 a^4 b^{10}}\)[/tex]
- [tex]\(\frac{1}{36 a^4 b^{10}}\)[/tex]
- [tex]\(\frac{36 a^4 b^{10}}{2}\)[/tex]
The correct equivalent expression is:
[tex]\[
\boxed{\frac{1}{36 a^4 b^{10}}}
\][/tex]
