https://file.aiquickdraw.com/mathbot/file/17294717720442r1jp0st.jpg

The expression in the image is:

\[
\left( \frac{-2a^3b^2c^0}{3a^2b^3c^7} \right)^{-2}
\]

### Step-by-Step Simplification:

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

   \[
   \frac{-2a^3b^2c^0}{3a^2b^3c^7}
   \]

   - \(c^0 = 1\), so we can remove \(c^0\) from the expression:
   
     \[
     \frac{-2a^3b^2}{3a^2b^3c^7}
     \]

   - For the variable \(a\):
     \[
     \frac{a^3}{a^2} = a^{3-2} = a^1 = a
     \]

   - For the variable \(b\):
     \[
     \frac{b^2}{b^3} = b^{2-3} = b^{-1} = \frac{1}{b}
     \]

   - The term with \(c\) remains \(c^{-7}\) (since it's in the denominator).

   Now the simplified fraction is:
   \[
   \frac{-2a}{3bc^7}
   \]

2. **Apply the exponent of \(-2\):**

   The next step is to apply the power of \(-2\) to each part of the fraction:
   \[
   \left( \frac{-2a}{3bc^7} \right)^{-2}
   \]
   
   When raising a fraction to a negative power, we invert the fraction and apply the positive power:

   \[
   \left( \frac{-2a}{3bc^7} \right)^{-2} = \left( \frac{3bc^7}{-2a} \right)^2
   \]

3. **Square both the numerator and denominator:**

   - Numerator:
     \[
     (3bc^7)^2 = 3^2 \cdot b^2 \cdot (c^7)^2 = 9b^2c^{14}
     \]

   - Denominator:
     \[
     (-2a)^2 = (-2)^2 \cdot a^2 = 4a^2
     \]

   So the expression becomes:
   \[
   \frac{9b^2c^{14}}{4a^2}
   \]

### Final Answer:

\[
\frac{9b^2c^{14}}{4a^2}
\]

---

Would you like a detailed explanation of any part, or do you have any questions?

Here are 5 related questions for further practice:
1. How would the expression change if the power was \(-3\) instead of \(-2\)?
2. What happens if \(c\) was raised to \(c^2\) instead of \(c^0\) in the numerator?
3. What is the result if the expression is raised to a positive power instead of a negative one?
4. How do you handle a negative exponent on a fraction in general?
5. Can you apply the same process if the base contains multiple variables?

**Tip:** When simplifying expressions with exponents, always simplify the terms inside the parentheses first before applying the exponent outside.

Simplify the expression ( (-2a^3b^2c^0) / (3a^2b^3c^7) )^(-2)

Learn how to simplify the algebraic expression ( (-2a^3b^2c^0) / (3a^2b^3c^7) )^(-2) using exponent rules and simplification techniques. This problem covers key algebra concepts and is suitable for students in Grades 9-11.

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