The sum of the real values of $x$ for which the middle term in

Middle Term, $\left(\frac{n}{2}+1\right)^{\text {th }}$ term in the binomial

expansion of $\left(\frac{x^{3}}{3}+\frac{3}{x}\right)^{8}$ is,

$T_{4+1}={ }^{8} C_{4}\left(\frac{x^{3}}{3}\right)^{4}\left(\frac{3}{x}\right)^{4}=5670$

$\Rightarrow \quad \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2} \times x^{12-4}=5670$

$\Rightarrow \quad x^{8}=81$

$\Rightarrow \quad x^{8}-81=0$

$\therefore$ sum of all values of $x=$ sum of roots of equation

$\left(x^{8}-81=0\right)$