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To find the smallest possible value of the expression

\[
\frac{32}{a} + \frac{128}{b} + ab
\]

where \( a \) and \( b \) are positive real numbers, we can apply the Arithmetic Mean-Geometric Mean (AM-GM) inequality. 

The AM-GM inequality states that for non-negative numbers, the arithmetic mean is always greater than or equal to the geometric mean. For three terms, it can be applied as follows:

\[
\frac{32}{a} + \frac{128}{b} + ab \geq 3 \sqrt[3]{\frac{32}{a} \cdot \frac{128}{b} \cdot ab}
\]

1. **Compute the product under the cube root:**
   \[
   \frac{32}{a} \cdot \frac{128}{b} \cdot ab = 32 \cdot 128 = 4096
   \]

2. **Take the cube root:**
   \[
   \sqrt[3]{4096} = 16
   \]

3. **Apply the AM-GM inequality:**
   \[
   \frac{32}{a} + \frac{128}{b} + ab \geq 3 \cdot 16 = 48
   \]

Thus, the minimum possible value of \( \frac{32}{a} + \frac{128}{b} + ab \) is \( 48 \).

### Conclusion
The smallest possible value of the expression is \(\boxed{48}\), which matches option 4.

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Would you like further clarification on any step?

### Related Questions
1. How does the AM-GM inequality help in optimization problems?
2. Can this minimum be achieved for specific values of \(a\) and \(b\)?
3. Why is the cube root used in this AM-GM application?
4. How would the solution change if the expression involved different constants?
5. Could the AM-GM inequality be applied to more complex expressions?

### Tip
The AM-GM inequality is a powerful tool for minimizing or maximizing expressions involving products and sums, especially when symmetry is present.

Let a and b be positive real numbers. What is the smallest possible value of \( \frac{32}{a} + \frac{128}{b} + ab \)?
Options:
1) 18
2) 24
3) 36
4) 48

Discover the solution to finding the smallest possible value of \( \frac{32}{a} + \frac{128}{b} + ab \) using the AM-GM inequality. This optimization problem involves applying the Arithmetic Mean-Geometric Mean inequality to determine the minimum value, making it suitable for students in Grades 10-12.

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