The sum of the real values of `x` for which the middle term in the binomial expansion of `(x^3/3 + 3/x)^8` equals `5670` is :

Options

A. 8
B. 0
C. 4
D. 6

Explanation

To solve the problem, we need to find the sum of the real values of \( x \) for which the middle term in the binomial expansion of \( \left( \frac{x^3}{3} + \frac{3}{x} \right)^8 \) equals \( 5670 \).

### Step-by-Step Solution:

1. **Identify the Middle Term**:
The binomial expansion of \( (a + b)^n \) has \( n + 1 \) terms. For \( n = 8 \), there are \( 9 \) terms, and the middle term is the \( 5^{th} \) term (since \( \frac{9 + 1}{2} = 5 \)).

2. **Write the General Term**:
The general term \( T_k \) in the expansion of \( (a + b)^n \) is given by:
\[ T_k = \binom{n}{k-1} a^{n-(k-1)} b^{k-1} \]
For our case, \( a = \frac{x^3}{3} \), \( b = \frac{3}{x} \), and \( n = 8 \). Therefore, the \( 5^{th} \) term \( T_5 \) is:
\[ T_5 = \binom{8}{4} \left( \frac{x^3}{3} \right)^{8-4} \left( \frac{3}{x} \right)^{4} \]

3. **Calculate \( T_5 \)**:
\[ T_5 = \binom{8}{4} \left( \frac{x^3}{3} \right)^{4} \left( \frac{3}{x} \right)^{4} \]
Simplifying this:
\[ T_5 = \binom{8}{4} \left( \frac{x^{12}}{3^4} \right) \left( \frac{3^4}{x^4} \right) = \binom{8}{4} \frac{x^{12}}{3^4} \cdot \frac{3^4}{x^4} = \binom{8}{4} x^{8} \]

4. **Set the Equation**:
We are given that this middle term equals \( 5670 \):
\[ \binom{8}{4} x^{8} = 5670 \]

5. **Calculate \( \binom{8}{4} \)**:
\[ \binom{8}{4} = \frac{8!}{4! \cdot 4!} = 70 \]
Thus, we have:
\[ 70 x^{8} = 5670 \]

6. **Solve for \( x^{8} \)**:
\[ x^{8} = \frac{5670}{70} = 81 \]

7. **Find \( x \)**:
Taking the eighth root:
\[ x = 81^{1/8} \]
Since \( 81 = 3^4 \), we have:
\[ x = 3^{4/8} = 3^{1/2} = \sqrt{3} \]
The other value is \( x = -\sqrt{3} \).

8. **Sum of Real Values**:
The sum of the real values of \( x \) is:
\[ \sqrt{3} + (-\sqrt{3}) = 0 \]

### Final Answer:
The sum of the real values of \( x \) is \( \boxed{0} \).