The problem asks for the simplified form of the expression:

$\left(\frac{a^{-3}b^{-2}c}{a^{-5}b^{-7}c^3}\right)^{-2}$

Step 1: Simplify the expression inside the parentheses

First, simplify the fraction by subtracting the exponents of the same bases:

$\frac{a^{-3}b^{-2}c^1}{a^{-5}b^{-7}c^3} = a^{-3 - (-5)}b^{-2 - (-7)}c^{1 - 3}$

This simplifies to:

$a^{2}b^{5}c^{-2}$

Step 2: Apply the exponent of -2

Now, raise the simplified expression to the power of -2:

$\left(a^{2}b^{5}c^{-2}\right)^{-2} = a^{2 \times (-2)}b^{5 \times (-2)}c^{-2 \times (-2)}$

This gives:

$a^{-4}b^{-10}c^{4}$

Final answer:

The simplified expression is $a^{-4}b^{-10}c^{4}$.

Thus, the correct answer is D.

Would you like more details or have any other questions? Here are five related questions to explore:

1. How do we simplify expressions with negative exponents?
2. How are exponents manipulated when dividing terms with the same base?
3. What are the rules for distributing exponents over multiplication and division?
4. How do you handle fractional exponents?
5. What happens when you raise a power to another power in exponents?

Tip: Always remember to simplify the base before applying the power, especially when negative exponents are involved.