Let's simplify the expression step by step:

The given expression is:

$\left(\frac{3a^{6}b^{6}}{5a^{5}b^{2}}\right)^{4}$

Step 1: Simplify the fraction inside the parentheses

$\frac{3a^6b^6}{5a^5b^2} = \frac{3}{5} \cdot a^{6-5} \cdot b^{6-2} = \frac{3}{5} \cdot a^1 \cdot b^4 = \frac{3a b^4}{5}$

So, the expression simplifies to: $\left(\frac{3ab^4}{5}\right)^{4}$

Step 2: Apply the exponent to both the numerator and denominator

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

Step 3: Simplify both the numerator and denominator

• For the numerator: $(3ab^4)^4 = 3^4 \cdot a^4 \cdot (b^4)^4 = 81a^4b^{16}$
• For the denominator: $5^4 = 625$

Final Simplified Expression:

$\frac{81a^4b^{16}}{625}$

So, the simplified expression is: $\frac{81a^4b^{16}}{625}$

Would you like further details on any step or have any questions?

Here are 5 related questions to expand on this:

1. How would you simplify an expression with negative exponents?
2. How do you apply exponent rules to fractional exponents?
3. What happens when you raise a product of variables to a power?
4. Can you simplify $\left(\frac{2x^3y^2}{4x^5}\right)^2$?
5. How do you simplify expressions involving radicals and exponents?

Tip: Always apply exponent rules to each variable separately when simplifying complex expressions.