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The problem asks to determine all possible values of \( n \in \mathbb{Z}^+ \) (positive integers) for which the expansion of 

\[
\left( 8x^3 - \frac{1}{2x} \right)^n
\]

has a non-zero constant term.

### Solution Outline:
We need to use the binomial expansion for terms of the form \( (a + b)^n \), which involves terms given by the general formula:

\[
T_k = \binom{n}{k} a^{n-k} b^k
\]

where \( a = 8x^3 \) and \( b = -\frac{1}{2x} \).

To find the constant term in the expansion, we focus on the powers of \( x \) in each term of the expansion. The total power of \( x \) in a term \( T_k \) is given by:

\[
\text{Power of } x = (3(n-k)) + (-1)(k)
\]

This simplifies to:

\[
\text{Power of } x = 3n - 4k
\]

For a term to be a constant, the power of \( x \) must be zero. Therefore, we solve:

\[
3n - 4k = 0
\]

Solving for \( k \), we get:

\[
k = \frac{3n}{4}
\]

For \( k \) to be an integer, \( 3n \) must be divisible by 4. This means that \( n \) must be a multiple of 4.

### Conclusion:
The possible values of \( n \) for which the expansion has a non-zero constant term are all multiples of 4, i.e.,

\[
n = 4, 8, 12, 16, \ldots
\]

Would you like more details on any step, or do you have any questions?

Here are 5 related questions to further explore:
1. How do you apply the binomial expansion to non-integer exponents?
2. What is the significance of the binomial coefficient in expansions like these?
3. How would the solution change if the term \( 8x^3 \) had a different power of \( x \)?
4. Can we use the same approach to find specific terms in the expansion (e.g., \( x^2 \) term)?
5. What are some common scenarios where binomial expansions appear in calculus?

**Tip:** Always consider the powers of variables in each term when looking for specific results (e.g., constant terms) in expansions.

Consider the expansion of (8x^3 - 1/(2x))^n where n ∈ Z^+. Determine all possible values of n for which the expansion has a non-zero constant term.

Explore how to find values of n such that the expansion of (8x^3 - 1/(2x))^n has a non-zero constant term. This solution involves binomial expansion, algebra, and understanding powers of polynomials. Ideal for Grade 11-12 students or early undergraduate levels.

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