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The problem asks to evaluate the expression \( p^3 - q^4 + \frac{3r}{2} \) given the values \( p = 6 \), \( q = 2 \), and \( r = 4 \).

Let’s solve this step by step.

1. **Substitute the values** into the expression:
   \[
   6^3 - 2^4 + \frac{3 \times 4}{2}
   \]

2. **Evaluate each term**:
   - \( 6^3 = 6 \times 6 \times 6 = 216 \)
   - \( 2^4 = 2 \times 2 \times 2 \times 2 = 16 \)
   - \( \frac{3 \times 4}{2} = \frac{12}{2} = 6 \)

3. **Now plug the results into the expression**:
   \[
   216 - 16 + 6
   \]

4. **Perform the operations**:
   \[
   216 - 16 = 200
   \]
   \[
   200 + 6 = 206
   \]

Thus, the value of the expression is **206**.

Would you like more details on any part of this solution? 

Here are 5 related questions to expand on this topic:
1. How would the result change if \( r \) were a fraction instead of an integer?
2. What happens if one of the variables is negative in this expression?
3. How do exponent rules apply when evaluating powers like \( p^3 \) or \( q^4 \)?
4. What if the division involved was more complex, like \( \frac{3r^2}{2r} \)?
5. How can we simplify complex algebraic expressions using factorization?

**Tip**: Always substitute the values carefully and solve exponents and multiplication before moving on to addition and subtraction according to the order of operations (PEMDAS).

Evaluate the expression p^3 − q^4 + 3r ÷ 2 for p = 6, q = 2, and r = 4.

Learn how to evaluate the algebraic expression p^3 − q^4 + 3r ÷ 2 by substituting the given values p = 6, q = 2, and r = 4. Step-by-step solution provided, covering exponent rules and order of operations. Suitable for Grades 6-8.

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