Let's simplify the expression \((p^3 q^4)^3 (pq)^2\).

### Step 1: Simplify each part of the expression.

- For \((p^3 q^4)^3\):
  \[
  (p^3 q^4)^3 = p^{3 \times 3} q^{4 \times 3} = p^9 q^{12}
  \]

- For \((pq)^2\):
  \[
  (pq)^2 = p^2 q^2
  \]

### Step 2: Multiply the simplified expressions together.

\[
p^9 q^{12} \times p^2 q^2
\]

### Step 3: Combine the like terms.

- For the \(p\) terms:
  \[
  p^9 \times p^2 = p^{9+2} = p^{11}
  \]

- For the \(q\) terms:
  \[
  q^{12} \times q^2 = q^{12+2} = q^{14}
  \]

### Final Simplified Expression:

\[
p^{11} q^{14}
\]

Would you like more details or have any questions? 

Here are some related questions:
1. How would the expression change if the exponents in \((pq)^2\) were different?
2. What is the general rule for simplifying \((a^m b^n)^p\)?
3. How does this expression simplify if both \(p\) and \(q\) are negative?
4. What happens if there is a fractional exponent in \((pq)^2\)?
5. How would the process change if there were coefficients in front of \(p\) and \(q\)?

**Tip:** When multiplying powers with the same base, add the exponents together.

Explore how to simplify the expression (p³ q⁴)³ (pq)² with detailed steps and explanations. This problem covers exponentiation and multiplication of powers, suitable for advanced high school students.

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