Question

The term independent of $x$ in the expansion of
$\left[\frac{x+1}{x^{2 / 3}-x^{1 / 3}+1}-\frac{x-1}{x-x^{1 / 2}}\right]^{10}, x \neq 1$, is equal to ______.

Answer 1

$\left(\left(x^{1 / 3}+1\right)-\left(\frac{\sqrt{x}+1}{\sqrt{x}}\right)\right)^{10}$
$\left(x^{1 / 3}-x^{-1 / 2}\right)^{10}$
$T_{r+1}=\,{}^{10}C_{r}\left(x^{1 / 3}\right)^{10-r}\left(-x^{-1 / 2}\right)^{r}$
$\frac{10-r}{3}-\frac{r}{2}=0 \Rightarrow 20-2 r-3 r=0$
$\Rightarrow r=4$
$T_{5}={ }^{10} C_{4}=\frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1}=210$