1. **State the problem:** Simplify the expression $\left(5x^{4}y^{-3}\right)^{-2}$. 2. **Recall the exponent rules:** - Power of a product: $\left(ab\right)^n = a^n b^n$ - Power of a power: $\left(a^m\right)^n = a^{mn}$ - Negative exponent: $a^{-n} = \frac{1}{a^n}$ 3. **Apply the power of a product rule:** $$\left(5x^{4}y^{-3}\right)^{-2} = 5^{-2} \cdot \left(x^{4}\right)^{-2} \cdot \left(y^{-3}\right)^{-2}$$ 4. **Apply the power of a power rule:** $$= 5^{-2} \cdot x^{4 \times (-2)} \cdot y^{-3 \times (-2)} = 5^{-2} \cdot x^{-8} \cdot y^{6}$$ 5. **Rewrite negative exponents as fractions:** $$= \frac{y^{6}}{5^{2} \cdot x^{8}} = \frac{y^{6}}{25x^{8}}$$ 6. **Final simplified expression:** $$\boxed{\frac{y^{6}}{25x^{8}}}$$ 7. **Select the correct answer:** This matches option **c**.