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The problem in the image asks to find the middle term of the expansion of \( (3x - \frac{1}{2x})^8 \). 

This is a binomial expansion problem, where we will use the **binomial theorem** to expand the expression. The general term in the expansion of \( (a + b)^n \) is given by:

\[
T_{r+1} = \binom{n}{r} a^{n-r} b^r
\]

For the given expression \( (3x - \frac{1}{2x})^8 \), we can identify:
- \( a = 3x \)
- \( b = -\frac{1}{2x} \)
- \( n = 8 \)

We are asked to find the middle term. In a binomial expansion, if \( n \) is even, the middle term is the \( \left(\frac{n}{2} + 1\right) \)-th term. Since \( n = 8 \), the middle term will be the \( \left(\frac{8}{2} + 1\right) = 5 \)-th term.

Let’s calculate the 5th term using the general term formula.

The 5th term corresponds to \( r = 4 \):

\[
T_5 = \binom{8}{4} (3x)^{8-4} \left(-\frac{1}{2x}\right)^4
\]

Now, let me calculate the value for you.The 5th term in the expansion of \( (3x - \frac{1}{2x})^8 \) is:

\[
T_5 = 354.375 \cdot x^0
\]

Since \( x^0 = 1 \), the middle term is simply:

\[
T_5 = 354.375
\]

Would you like a detailed explanation of the steps, or do you have any further questions?

Here are 5 related questions:
1. How is the binomial theorem applied to expressions with both positive and negative terms?
2. What is the general formula for finding a specific term in a binomial expansion?
3. How can you find the middle term if the power of the binomial is odd?
4. What are some applications of the binomial theorem in calculus or probability?
5. How do coefficients of terms in a binomial expansion change as the power increases?

**Tip**: In binomial expansions, the middle term often provides symmetry, and powers of \(x\) play a crucial role in understanding behavior at certain points (like peaks).

Find the middle term of the expansion (3x - 1/(2x))^8

Explore the process of finding the middle term in the binomial expansion of (3x - 1/(2x))^8 using the binomial theorem. This detailed guide is suitable for students in Grades 10-12, covering key concepts in algebra and binomial expansions.

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