$\text{ The term independent of }x\text{ in the expansion of }{\left[\frac{x+1}{x^{2/3}-x^{1/3}+1}-\frac{x-1}{x-x^{1/2}}\right]}^{10},x\ne 1\text{ is }$ =

$\begin{array}{l}\text{The given binomial expansion is }{\left(\frac{x+1}{x^{\frac{2}{3}}-x^{\frac{1}{3}}+1}-\frac{x-1}{x-x^{\frac{1}{2}}}\right)}^{10}\\ \text{This can be simplify as }{\left(\frac{\left(x^{\frac{1}{3}}+1\right)\left(x^{\frac{2}{3}}-x^{\frac{1}{3}}+1\right)}{\left(x^{\frac{2}{3}}-x^{\frac{1}{3}}+1\right)}-\frac{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}{\sqrt{x}\left(\sqrt{x}-1\right)}\right)}^{10}={\left(x^{\frac{1}{3}}+1-1-x^{-\frac{1}{2}}\right)}^{10}={\left(x^{\frac{1}{3}}-x^{-\frac{1}{2}}\right)}^{10}\end{array}$

$r=\frac{np}{p+q}=\frac{10\left({\displaystyle \frac{1}{3}}\right)}{{\displaystyle \frac{1}{3}}+{\displaystyle \frac{1}{2}}}=4$

$\text{ To get independent term }\mathrm{r}=4 \therefore T_5={}^{10}C_4=210$