Let's simplify the expression \(\frac{(6a^2b^3)^2}{(3ab^2)^3}\).

### Step 1: Expand the powers in the numerator and the denominator
- **Numerator**: \((6a^2b^3)^2\)
  \[
  = 6^2 \cdot (a^2)^2 \cdot (b^3)^2 = 36a^4b^6
  \]
- **Denominator**: \((3ab^2)^3\)
  \[
  = 3^3 \cdot (a)^3 \cdot (b^2)^3 = 27a^3b^6
  \]

### Step 2: Simplify the fraction
\[
\frac{36a^4b^6}{27a^3b^6}
\]

### Step 3: Simplify each part
- **Coefficients**: \(\frac{36}{27} = \frac{4}{3}\)
- **\(a\)-terms**: \(\frac{a^4}{a^3} = a^{4-3} = a\)
- **\(b\)-terms**: \(\frac{b^6}{b^6} = 1\)

### Final Answer
\[
\frac{4a}{3}
\]

Would you like any further details or have any questions?

Here are some related questions you might find interesting:
1. How do you simplify \(\frac{(4x^3y^2)^3}{(2x^2y)^4}\)?
2. What is the result of \(\frac{(2a^2b)^4}{(4ab^3)^2}\)?
3. How does the power rule apply to exponents in a fraction?
4. Can you explain the process of simplifying \(\frac{(5x^2y)^3}{(10xy^2)^2}\)?
5. What happens when the bases in the numerator and denominator are the same?
6. How would you handle an expression like \(\frac{(2p^3q)^5}{(4p^2q^4)^2}\)?
7. What is the simplified form of \(\frac{(7a^2b^4)^2}{(2ab^3)^3}\)?
8. How do you simplify an expression when the exponents are negative?

**Tip**: When simplifying expressions involving exponents, always expand and simplify each part step-by-step to avoid errors.

Learn how to simplify the expression (6a^2b^3)^2/(3ab^2)^3 with a step-by-step solution. This problem involves exponentiation and fraction simplification, suitable for high school students.

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