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Let's simplify the given expression: \((3x^3 y^2)^2\).

### Step-by-step solution:

1. **Apply the exponent to the constant**:  
   \[
   3^2 = 9
   \]

2. **Apply the exponent to each term inside the parentheses**:  
   For \(x^3\), apply the exponent:
   \[
   (x^3)^2 = x^{3 \times 2} = x^6
   \]
   For \(y^2\), apply the exponent:
   \[
   (y^2)^2 = y^{2 \times 2} = y^4
   \]

3. **Combine all the terms**:
   \[
   9x^6 y^4
   \]

Thus, the simplified expression is \(9x^6 y^4\).

The correct answer is the first option: **\(9x^6 y^4\)**.

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Would you like any more details or have any questions?

### Related questions:
1. How do you simplify expressions with multiple variables and exponents?
2. What is the general rule for applying exponents to powers?
3. Can you show how to simplify a more complex polynomial expression?
4. How would the result change if the expression had a negative exponent?
5. What if the base number in the expression was a fraction?

**Tip**: When applying an exponent to a product inside parentheses, remember to apply the exponent to each factor separately, including numbers and variables.

Learn how to simplify (3x^3 y^2)^2 with this detailed algebra solution. This problem covers key algebraic concepts, including the power of a power rule, multiplying exponents, and combining terms. Suitable for students in Grades 8-10.

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