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You have uploaded a mathematical expression, which appears as:

\[
\left( \frac{-7a^2b^3c^0}{3a^3b^4c^3} \right)^{-4}
\]

Let's break it down and simplify:

### Step 1: Simplify the expression inside the parentheses
The given expression is:
\[
\frac{-7a^2b^3c^0}{3a^3b^4c^3}
\]

- \( c^0 = 1 \), so we can eliminate \( c^0 \).
- Combine like terms:
  - For \( a \): \( a^2 \) in the numerator and \( a^3 \) in the denominator gives \( a^{2-3} = a^{-1} \).
  - For \( b \): \( b^3 \) in the numerator and \( b^4 \) in the denominator gives \( b^{3-4} = b^{-1} \).
  - \( c^3 \) remains in the denominator.

So, the simplified expression inside the parentheses is:
\[
\frac{-7a^{-1}b^{-1}}{3c^3}
\]

Which can be rewritten as:
\[
\frac{-7}{3} \cdot a^{-1} \cdot b^{-1} \cdot c^{-3}
\]

### Step 2: Apply the exponent \( -4 \)
Now we need to apply the \( -4 \) exponent to the entire expression:
\[
\left( \frac{-7}{3} \cdot a^{-1} \cdot b^{-1} \cdot c^{-3} \right)^{-4}
\]

- Raise each part to the power of \( -4 \):
  \[
  \left( \frac{-7}{3} \right)^{-4} \cdot (a^{-1})^{-4} \cdot (b^{-1})^{-4} \cdot (c^{-3})^{-4}
  \]

This simplifies to:
- \( \left( \frac{-7}{3} \right)^{-4} = \left( \frac{3}{-7} \right)^4 = \frac{3^4}{(-7)^4} = \frac{81}{2401} \)
- \( (a^{-1})^{-4} = a^4 \)
- \( (b^{-1})^{-4} = b^4 \)
- \( (c^{-3})^{-4} = c^{12} \)

### Step 3: Final simplified expression
Combining everything, we get:
\[
\frac{81}{2401} \cdot a^4 \cdot b^4 \cdot c^{12}
\]

Thus, the simplified expression is:
\[
\frac{81a^4b^4c^{12}}{2401}
\]

Would you like further details or have any questions?

Here are 5 related questions you could explore next:
1. How does the exponent rule work when simplifying terms with negative exponents?
2. What happens when the base of a fraction is negative, and it's raised to an even power?
3. Can this expression be used in a specific algebraic context, like solving equations?
4. How can we extend this simplification to larger exponents or more complex expressions?
5. What is the impact of applying different powers to a rational expression like this one?

**Tip:** Remember, when raising a fraction to a negative power, you can invert the fraction and apply the positive power.

Learn how to simplify the algebraic expression ( (-7a^2 b^3 c^0) / (3a^3 b^4 c^3) )^-4. This problem involves simplifying exponents, negative exponents, and algebraic fractions. Detailed step-by-step solution provided.

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