Explanation
\left\{\frac{(x+1)}{(x^{2/3}+1-x^{1/3})}-\frac{(x+1)}{(x-x^{1/2})}\right\}^{10}, x>1
=[x3+1(3x)3+13−x3(x−1)(3x)2−12]10
\begin{aligned}
\therefore\quad(x^{\frac{1}{3}})^{3}+1^{3} & =(x^{\frac{1}{3}}+1)(x^{\frac{2}{3}}+1-x^{\frac{1}{3}}) \\
(\sqrt{x})^{2}-1 & =(\sqrt{x}+1)(\sqrt{x}-1) \\
& \Rightarrow\left(\frac{(x+1)}{x^{\frac{2}{3}}+1-x^{\frac{1}{3}}}-\frac{(x-1)}{\left(x-x^{\frac{1}{2}}\right)}\right) \\
& & & =\left((x^{\frac{1}{3}}+1)-\left(\frac{\sqrt{x}+1}{\sqrt{x}}\right)\right)^{10} \\
& & & =\left(x^{\frac{1}{3}}-\frac{1}{\sqrt{x}}\right)^{10} \\
\mathrm{T}_{r+1} & ={}^{10}C_{r}(x)^{\frac{10-r}{3}}(-1)^{r}(x)^{-\frac{r}{2}} \\
& & & ={}^{10}C_{r}(-1)^{r}(x)^{\frac{10-r}{3}-\frac{r}{2}}
\end{aligned}
Term independent of x=310−r−2r=0
(20−2r)−3r=0
r=4
⇒10C4(−1)4=210
