This expression seems to be a product of several terms involving powers and fractions. Let's break it down step-by-step.

The given expression is: $\left( \frac{5}{9} \right)^{-2} \times \left( \frac{-3}{5} \right)^{-3} \times \left( \frac{3}{5} \right)^0$

Step 1: Simplify each term individually.

1. $\left( \frac{5}{9} \right)^{-2}$: Using the property $a^{-n} = \frac{1}{a^n}$, we get: $\left( \frac{5}{9} \right)^{-2} = \left( \frac{9}{5} \right)^2 = \frac{81}{25}$

2. $\left( \frac{-3}{5} \right)^{-3}$: Similarly, applying the same rule: $\left( \frac{-3}{5} \right)^{-3} = \left( \frac{-5}{3} \right)^3 = \frac{-125}{27}$

3. $\left( \frac{3}{5} \right)^0$: Any non-zero number raised to the power of zero is 1: $\left( \frac{3}{5} \right)^0 = 1$

Step 2: Multiply the simplified terms together.

Now, we combine all the simplified terms: $\frac{81}{25} \times \frac{-125}{27} \times 1$

Simplify the multiplication: $= \frac{81 \times -125}{25 \times 27} = \frac{-10125}{675}$

Step 3: Simplify the final fraction.

To simplify $\frac{-10125}{675}$, divide both the numerator and the denominator by their greatest common divisor, which is 225: $\frac{-10125 \div 225}{675 \div 225} = \frac{-45}{3}$

Finally, simplify: $\frac{-45}{3} = -15$

Final Answer:

The value of the expression is $-15$.

Would you like further details on how the steps were performed, or do you have any additional questions? Here are a few related questions to explore:

1. How do we apply negative exponents in fractions?
2. Can you explain the properties of exponents in more detail?
3. What happens if we raise a fraction to the power of zero?
4. How do you simplify large fractions when multiplying and dividing?
5. What is the greatest common divisor (GCD), and how do you calculate it?

Tip: Always double-check the signs when working with negative numbers in exponentiation!