To solve $\left( -\frac{3}{5} \right)^{-4}$, we use the following rules of exponents:

1. A negative exponent means we take the reciprocal of the base and make the exponent positive: $\left( -\frac{3}{5} \right)^{-4} = \left( \frac{5}{-3} \right)^4$

2. Now, raise both the numerator and the denominator to the power of 4: $\left( \frac{5}{-3} \right)^4 = \frac{5^4}{(-3)^4}$

3. Calculate the powers: $5^4 = 625 \quad \text{and} \quad (-3)^4 = 81$

4. Therefore: $\left( -\frac{3}{5} \right)^{-4} = \frac{625}{81}$

So, $\left( -\frac{3}{5} \right)^{-4} = \frac{625}{81}$.

Would you like more details or have any questions?

Here are 5 related questions to explore:

1. What happens when the exponent is positive for fractions with negative bases?
2. How do you simplify $\left( \frac{2}{7} \right)^{-3}$?
3. Can you explain why any number raised to the 0th power equals 1?
4. What is the result of $\left( -\frac{4}{9} \right)^3$?
5. How does changing the sign of the exponent affect the result?

Tip: When raising a fraction to a negative exponent, always take the reciprocal before applying the power.