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

${\left(\frac{x^3}{3}+\frac{3}{x}\right)}^8$ equal 5670 is

Given binomial expansion is ${\left(\frac{x^3}{3}+\frac{3}{x}\right)}^8$So, middle term $T_5{=}^8{C}_4{\left(\frac{x^3}{3}\right)}^4\cdot {\left(\frac{3}{x}\right)}^4=5670$

$\begin{array}{l}{\Rightarrow }^8{C}_4\frac{x^{12}}{3^4}\cdot \frac{3^4}{x^4}=5670\\ \Rightarrow 70x^8=5670\Rightarrow x^8=81\Rightarrow x=\pm \sqrt{3}\end{array}$

$\therefore$ Sum of real values of x is 0.