The term independent of x in expansion of ${\left(\frac{x+1}{x^{2/3}-x^{1/3}+1}-\frac{x-1}{x-x^{1/2}}\right)}^{10}$ is 

The Given binomial expansion is ${\left(\frac{x+1}{x^{2/3}-x^{1/3}+1}-\frac{x-1}{x-x^{1/2}}\right)}^{10}$

The above expression can be written as 

${\left(\frac{\left(x^{{\displaystyle \frac{1}{3}}}+1\right)\left(x^{2/3}-x^{1/3}+1\right)}{x^{2/3}-x^{1/3}+1}-\frac{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}{\sqrt{x}\left(\sqrt{x}-1\right)}\right)}^{10}$

$={\left((x^{1/3}+1)-\left(\frac{\sqrt{x}+1}{\sqrt{x}}\right)\right)}^{10}$

$={\left(x^{1/3}+1-1-\frac{1}{\sqrt{x}}\right)}^{10}$

$={(x^{1/3}-x^{-1/2})}^{10}$

The term independent of $x$ in the expansion ${\left(p+q\right)}^n$is $T_{r+1}$ and $r=\frac{np}{p+q},r=4$

So required term $T_5=210$