Question

The term independent of $x$ in the expansion of ${(\frac{x+1}{x^{2/3}-x^{1/3}+1}-\frac{x-1}{x-x^{1/2}})}^{10}$, where $x\ne 0,1$ is equal to

Answer 1

${(\frac{x+1}{x^{2/3}-x^{1/3}+1}-\frac{x-1}{x-x^{1/2}})}^{10}={(\frac{x+1}{x^{2/3}-x^{1/3}+1}\times \frac{1+x^{1/3}}{1+x^{1/3}}-\frac{x-1}{x-x^{1/2}}\times \frac{x+x^{1/2}}{x+x^{1/2}})}^{10}={((x^{1/3}+1)-\frac{\sqrt{x}+1}{\sqrt{x}})}^{10}={(x^{1/3}-x^{-1/2})}^{10}$

$T_{r+1}={}^{10}{C}_r\cdot x^{\frac{10-r}{3}}\cdot (-x{)}^{-\frac{r}{2}}$
$\because \frac{10-r}{3}=\frac{r}{2}\Rightarrow r=4$
put $r=4,T_5={}^{10}{C}_4=210$