The mathematical expression in the image is:

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

Simplifying Step-by-Step:

1. Simplify Inside the Parentheses:

   • $c^0 = 1$ (anything raised to the power of 0 is 1).
   • So, we simplify: $\frac{7a^2 b^3}{3a^3 b^4 c^3}$

2. Divide the Like Terms:

   • For $a$'s: $\frac{a^2}{a^3} = a^{2-3} = a^{-1}$.
   • For $b$'s: $\frac{b^3}{b^4} = b^{3-4} = b^{-1}$.
   • For $c$'s: $\frac{1}{c^3} = c^{-3}$.

   So the expression simplifies to: $\frac{7}{3} a^{-1} b^{-1} c^{-3}$

3. Apply the Exponent Outside the Parentheses: The entire expression is raised to the power of $-4$: $\left( \frac{7}{3} a^{-1} b^{-1} c^{-3} \right)^{-4}$

4. Distribute the Exponent:

   • $\left( \frac{7}{3} \right)^{-4} = \frac{3^4}{7^4} = \frac{81}{2401}$.
   • $(a^{-1})^{-4} = a^4$.
   • $(b^{-1})^{-4} = b^4$.
   • $(c^{-3})^{-4} = c^{12}$.

   So, the expression becomes: $\frac{81}{2401} a^4 b^4 c^{12}$

Final Answer:

$\boxed{\frac{81 a^4 b^4 c^{12}}{2401}}$

Let me know if you would like further details or have any questions!

Related Questions:

1. How do you simplify negative exponents when they appear in a denominator?
2. What is the rule for dividing terms with the same base but different exponents?
3. How do you handle exponents when raising fractions to a power?
4. Can you explain why anything raised to the power of zero equals one?
5. How does raising a term with a negative exponent to a power affect its sign?

Tip:

When simplifying fractions with exponents, remember to subtract the exponents of like bases (numerator minus denominator).