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We are given the following expression:

\[
\left( \frac{(ab^2c)^{-3}}{b^{-3}} \right)^2
\]

### Step-by-Step Solution:

1. **Simplify the inside of the brackets:**

   Let's start by simplifying the fraction inside the parentheses:
   \[
   \frac{(ab^2c)^{-3}}{b^{-3}}
   \]

   Use the property of exponents \(\frac{a^m}{a^n} = a^{m-n}\) to simplify the powers of \(b\):
   \[
   \frac{(ab^2c)^{-3}}{b^{-3}} = (ab^2c)^{-3} \cdot b^{3}
   \]
   This is because \(\frac{1}{b^{-3}} = b^3\).

2. **Distribute the exponents:**

   Now apply the exponent of \(-3\) to each factor inside the parentheses:
   \[
   (ab^2c)^{-3} = a^{-3} \cdot (b^2)^{-3} \cdot c^{-3} = a^{-3} \cdot b^{-6} \cdot c^{-3}
   \]
   So we now have:
   \[
   a^{-3}b^{-6}c^{-3} \cdot b^{3}
   \]

3. **Simplify further:**

   Combine the powers of \(b\) using the exponent rules:
   \[
   a^{-3}b^{-6+3}c^{-3} = a^{-3}b^{-3}c^{-3}
   \]

4. **Square the result:**

   Now, square the entire expression as indicated by the outer exponent of 2:
   \[
   \left( a^{-3}b^{-3}c^{-3} \right)^2 = a^{-6}b^{-6}c^{-6}
   \]

### Final Answer:
\[
a^{-6}b^{-6}c^{-6}
\]

This is the simplified form of the given expression.

Would you like more details on any step or have any questions?

### Related Questions:

1. What are the properties of exponents and how can they simplify complex expressions?
2. How do you handle negative exponents when simplifying algebraic expressions?
3. What happens when you multiply exponents with the same base?
4. Can you explain how distributing exponents works in a more general form?
5. How would this problem change if different exponents were used?

### Tip:
When working with negative exponents, always remember that they represent the reciprocal of the base raised to the positive exponent!

Simplify the expression ((ab^2c)^(-3) / b^(-3))^2

Learn how to simplify the algebraic expression ((ab^2c)^(-3) / b^(-3))^2 by applying the laws of exponents. This detailed step-by-step solution is suitable for high school students studying exponents and algebra.

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