Question

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

Answer 1

Answer: 210

${\left(\left(x^{1/3}+1\right)-\left(\frac{\sqrt{x}+1}{\sqrt{x}}\right)\right)}^{10}$
${\left(x^{1/3}-x^{-1/2}\right)}^{10}$
$T_{r+1}={}^{10}{C}_r{\left(x^{1/3}\right)}^{10-r}{\left(-x^{-1/2}\right)}^r$
$\frac{10-r}{3}-\frac{r}{2}=0\Rightarrow 20-2r-3r=0$
$\Rightarrow r=4$
$T_5={}^{10}{C}_4=\frac{10\times 9\times 8\times 7}{4\times 3\times 2\times 1}=210$