The given expression is:

[ \frac{\left[\left(\frac{3}{5}\right)^3 \cdot \left(\frac{3}{5}\right)^3 \right]^4}{\left[\left(\frac{3}{5}\right)^2 \right]^2} ]

Step-by-Step Solution:

1. Simplify the numerator: The expression inside the brackets in the numerator is $\left(\frac{3}{5}\right)^3 \cdot \left(\frac{3}{5}\right)^3$. Using the property of exponents where $a^m \cdot a^n = a^{m+n}$:

   $\left(\frac{3}{5}\right)^3 \cdot \left(\frac{3}{5}\right)^3 = \left(\frac{3}{5}\right)^{3+3} = \left(\frac{3}{5}\right)^6$

   Now the numerator becomes: $\left[\left(\frac{3}{5}\right)^6\right]^4$

2. Apply the power of a power rule: $\left(\frac{3}{5}\right)^{6 \cdot 4} = \left(\frac{3}{5}\right)^{24}$

3. Simplify the denominator: The denominator is $\left[\left(\frac{3}{5}\right)^2\right]^2$. Again, using the power of a power rule:

   $\left(\frac{3}{5}\right)^{2 \cdot 2} = \left(\frac{3}{5}\right)^4$

4. Now the expression becomes: $\frac{\left(\frac{3}{5}\right)^{24}}{\left(\frac{3}{5}\right)^4}$

5. Simplify using the quotient rule of exponents: $\left(\frac{3}{5}\right)^{24 - 4} = \left(\frac{3}{5}\right)^{20}$

Thus, the simplified expression is:

$\left(\frac{3}{5}\right)^{20}$

Would you like further details or have any questions?

Related Questions:

1. How does the quotient rule for exponents work in simplifying expressions?
2. Can you explain how to apply the power of a product rule to similar problems?
3. What happens if the bases in the numerator and denominator are different?
4. How does this problem change if negative exponents are introduced?
5. How would you simplify if the exponent in the numerator or denominator were a fraction?

Tip: Always remember to apply the rules of exponents carefully, especially when dealing with multiple layers of powers and products.