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

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.

This page provides a detailed step-by-step solution to the expression \left(\frac{3a^{6}b^{6}}{5a^{5}b^{2}}\right)^{4}. Learn how to simplify using exponent rules and fraction simplification, including key algebraic formulas and concepts. Suitable for Grades 9-11 students.

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