https://file.aiquickdraw.com/mathbot/file/1727089126311gnoup841.jpg

Let's solve the given expression from the image:

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

### Step 1: Simplify the inner expression
We can simplify the terms inside the parentheses first.

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

1. **Simplify the constants**:
   - The constant is \(3\), so no simplification is needed here.

2. **Simplify the powers of \(a\)**:
   - In the numerator, the power of \(a\) is \(2\).
   - In the denominator, the power of \(a\) is \(1\).
   - Subtract the exponents: \(a^{2-1} = a^1\).

3. **Simplify the powers of \(b\)**:
   - In the numerator, the power of \(b\) is \(1\).
   - In the denominator, the power of \(b\) is \(-2\).
   - Subtract the exponents: \(b^{1 - (-2)} = b^{1 + 2} = b^3\).

4. **Simplify the powers of \(c\)**:
   - In the numerator, the power of \(c\) is \(-3\).
   - In the denominator, the power of \(c\) is \(2\).
   - Subtract the exponents: \(c^{-3 - 2} = c^{-5}\).

Thus, the simplified expression inside the parentheses is:

\[
3a b^3 c^{-5}
\]

### Step 2: Square the entire expression

Now we square the simplified expression:

\[
\left( 3a b^3 c^{-5} \right)^2
\]

1. **Square the constant**: 
   \(3^2 = 9\).

2. **Square the powers of \(a\)**:
   \(a^1 \rightarrow (a^1)^2 = a^2\).

3. **Square the powers of \(b\)**:
   \(b^3 \rightarrow (b^3)^2 = b^6\).

4. **Square the powers of \(c\)**:
   \(c^{-5} \rightarrow (c^{-5})^2 = c^{-10}\).

Thus, the final expression is:

\[
9a^2 b^6 c^{-10}
\]

### Final Answers:
- **Power of \(a\)**: \(2\)
- **Power of \(b\)**: \(6\)
- **Power of \(c\)**: \(-10\), but since the answer requires positive powers, we rewrite this as \( \frac{1}{c^{10}} \).
- **Constant value**: \(9\)
- **Positions**:
  - \(a\) is in the numerator.
  - \(b\) is in the numerator.
  - \(c\) is in the denominator (because of the negative power).
  - Constant value is in the numerator.

Let me know if you want further clarifications or details. Here are five related questions to consider:

1. How do you apply the power rule for exponents when simplifying fractions?
2. What happens when you raise a fraction to a power?
3. How do you convert negative exponents into fractions?
4. Can you simplify a similar expression where variables are raised to fractional powers?
5. How would the solution change if the original expression involved cube roots?

### Tip:
When working with negative exponents, remember that they indicate reciprocals (e.g., \(x^{-n} = \frac{1}{x^n}\)).

Simplify (3a^2 b c^{-3} / ab^{-2} c^2)^2 as much as possible.

Learn how to simplify the expression (3a^2 b c^{-3} / ab^{-2} c^2)^2 using exponent rules and algebraic principles. This problem involves understanding negative exponents, simplifying fractions, and applying the power of a quotient rule. Suitable for Grades 8-10 students.

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