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The given expression is:

\[
\left( \frac{x^2}{a^3} \right)^{-1} \cdot \left( \frac{3a^2}{4x^3} \right)^2 \cdot 5xa^{-4}
\]

Let’s simplify the expression step by step:

### Step 1: Apply the exponent to the first term
\[
\left( \frac{x^2}{a^3} \right)^{-1} = \frac{a^3}{x^2}
\]

### Step 2: Apply the exponent to the second term
\[
\left( \frac{3a^2}{4x^3} \right)^2 = \frac{(3a^2)^2}{(4x^3)^2} = \frac{9a^4}{16x^6}
\]

### Step 3: Combine all terms
Now we have:
\[
\frac{a^3}{x^2} \cdot \frac{9a^4}{16x^6} \cdot 5xa^{-4}
\]

### Step 4: Multiply the constants
\[
5 \cdot 9 = 45
\]
So the constant part becomes:
\[
\frac{45}{16}
\]

### Step 5: Combine powers of \( a \)
\[
a^3 \cdot a^4 \cdot a^{-4} = a^{3 + 4 - 4} = a^3
\]

### Step 6: Combine powers of \( x \)
\[
x^{-2} \cdot x^{-6} \cdot x = x^{-2 - 6 + 1} = x^{-7}
\]

### Final Expression
The simplified form of the expression is:
\[
\frac{45a^3}{16x^7}
\]

Would you like further details or have any questions? 

Here are five related questions to explore:
1. How do you handle negative exponents in an expression?
2. What is the process for multiplying terms with the same base?
3. How can expressions involving fractions and exponents be simplified?
4. What happens if exponents involve more complex terms like polynomials?
5. How does simplification of terms change if there are additional constants involved?

**Tip:** Always handle the exponents first in each part of the expression before multiplying terms together to avoid mistakes in simplification.

Simplify the following expression: (x^2 / a^3)^-1 * (3a^2 / 4x^3)^2 * 5x a^-4.

Learn how to simplify a complex algebraic expression involving exponents, fractions, and negative powers. This detailed solution covers key concepts such as the laws of exponents and multiplication of terms. Suitable for high school students in Grades 9-12.

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