To simplify the expression:

\[
\frac{(a^2b^3)^2}{c^3} \div \frac{a^3b^2c}{a^{-2}}
\]

we will follow the order of operations and apply the laws of exponents step by step.

### Step 1: Simplify the powers

First, simplify \((a^2b^3)^2\):

\[
(a^2b^3)^2 = (a^2)^2 \cdot (b^3)^2 = a^{2 \cdot 2} \cdot b^{3 \cdot 2} = a^4 \cdot b^6
\]

Now, rewrite the expression with these simplified powers:

\[
\frac{a^4b^6}{c^3} \div \frac{a^3b^2c}{a^{-2}}
\]

### Step 2: Rewrite the division as multiplication

Division by a fraction is the same as multiplication by its reciprocal:

\[
\frac{a^4b^6}{c^3} \times \frac{a^{-2}}{a^3b^2c}
\]

### Step 3: Combine fractions

Combine the fractions by multiplying the numerators and the denominators:

\[
\frac{a^4b^6 \cdot a^{-2}}{c^3 \cdot a^3b^2c}
\]

### Step 4: Simplify the exponents

Combine like terms by adding the exponents of the same bases:

\[
\text{For } a: \quad a^{4 + (-2) - 3} = a^{4 - 2 - 3} = a^{-1}
\]

\[
\text{For } b: \quad b^{6 - 2} = b^4
\]

\[
\text{For } c: \quad c^{-3 - 1} = c^{-4}
\]

Now, substitute these simplified exponents into the expression:

\[
a^{-1}b^4c^{-4}
\]

### Step 5: Rewrite with positive exponents

To express the answer with positive exponents, rewrite negative exponents as reciprocals:

\[
\frac{b^4}{a \cdot c^4}
\]

### Conclusion

The simplified expression is:

\[
\boxed{\frac{b^4}{ac^4}}
\]

Would you like more details or have any questions?

### Related Questions:

1. How do you simplify expressions with multiple variables and exponents?
2. What are the rules of exponents, and how are they applied in algebraic expressions?
3. How can expressions with negative exponents be rewritten with positive exponents?
4. What is the importance of the order of operations in simplifying algebraic expressions?
5. How do you simplify complex fractions involving multiple variables?

**Tip:** When dealing with exponents, remember that multiplying like bases involves adding their exponents, and dividing like bases involves subtracting their exponents.

Learn how to simplify the expression (a^2b^3)^2 / c^3 / a^3b^2c / a^-2 step-by-step with detailed explanations. This problem involves exponents, fractions, and the order of operations, suitable for high school level mathematics.

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