To simplify the given expression $\left(\frac{3}{5}\right)^4 \left(\frac{8}{5}\right)^{-12} \left(\frac{32}{5}\right)^6$, we follow these steps:

1. Combine the fractions using properties of exponents.
2. Simplify the resulting expression.

Here's the expression: $\left(\frac{3}{5}\right)^4 \left(\frac{8}{5}\right)^{-12} \left(\frac{32}{5}\right)^6$

Step 1: Combine the fractions: $\left(\frac{3}{5}\right)^4 = \frac{3^4}{5^4}$ $\left(\frac{8}{5}\right)^{-12} = \left(\frac{8^{-12}}{5^{-12}}\right) = \frac{5^{12}}{8^{12}}$ $\left(\frac{32}{5}\right)^6 = \frac{32^6}{5^6}$

Now, combine all these together: $\frac{3^4}{5^4} \cdot \frac{5^{12}}{8^{12}} \cdot \frac{32^6}{5^6}$

Step 2: Simplify the exponents: $\frac{3^4 \cdot 5^{12} \cdot 32^6}{5^4 \cdot 8^{12} \cdot 5^6}$

Combine the exponents of $5$: $\frac{3^4 \cdot 5^{12}}{5^{10}} = \frac{3^4 \cdot 5^{2}}{8^{12}} \cdot 32^6$

Next, simplify the powers of 32 and 8: $8 = 2^3 \implies 8^{12} = (2^3)^{12} = 2^{36}$ $32 = 2^5 \implies 32^6 = (2^5)^6 = 2^{30}$

Substitute these back: $\frac{3^4 \cdot 5^{2}}{2^{36}} \cdot 2^{30}$

Combine the exponents of 2: $\frac{3^4 \cdot 5^{2}}{2^{6}}$

Step 3: Final simplification: $\frac{81 \cdot 25}{64} = \frac{2025}{64}$

Thus, the simplified form of the expression is: $\frac{2025}{64}$

Would you like further details or have any other questions?

Here are 5 questions you might consider:

1. How do you handle negative exponents in fractions?
2. What are the properties of exponents used in simplifying expressions?
3. How can you simplify complex fractions step by step?
4. What is the process for combining exponents when multiplying fractions?
5. Can you explain the difference between base and exponent in exponential expressions?

Tip: Always break down complex expressions into smaller parts and simplify step by step, using exponent rules and combining like terms for clarity.