The given expression is \((6u^2 v^3)^2\). To simplify this, we will apply the power of a product rule, which states that \((ab)^n = a^n b^n\).

### Step-by-step solution:
1. First, apply the exponent to each term inside the parentheses:
\[
(6u^2 v^3)^2 = (6)^2 \cdot (u^2)^2 \cdot (v^3)^2
\]

2. Simplify each term:
\[
(6)^2 = 36, \quad (u^2)^2 = u^{2 \cdot 2} = u^4, \quad (v^3)^2 = v^{3 \cdot 2} = v^6
\]

3. Now, multiply everything together:
\[
36u^4v^6
\]

### Final Answer:
\[
36u^4v^6
\]

Would you like more details or have any questions?

Here are five related questions:
1. How do you simplify \((3x^2y^4)^3\)?
2. What is the result of \((2a^3b^5)^2\)?
3. How do you apply the power rule to \((5p^2q)^3\)?
4. What happens when you square a binomial expression like \((x + y)^2\)?
5. How do you simplify \((2x^3z)^4\)?

**Tip:** Always apply exponents to both constants and variables separately when simplifying expressions with multiple terms inside parentheses.

This problem shows how to simplify the algebraic expression (6u^2 v^3)^2 using the Power of a Product Rule. The solution applies exponent rules to both constants and variables, yielding the final result of 36u^4v^6. Suitable for students in Grades 8-10.

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