The term independent of $x$ in the expansion of $\left(\frac{(x+1)}{\left(x^{2 / 3}+1-x^{1 / 3}\right)}-\frac{(x+1)}{\left(x-x^{1 / 2}\right)}\right)^{10}, x\gt1$ is:

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Aicharya · Accepted Answer

$\begin{aligned} & \left(\frac{(x+1)}{\left(x^{\frac{2}{3}}+1-x^{\frac{1}{3}}ight)}-\frac{(x-1)}{\left(x-x^{\frac{1}{2}}ight)}ight)^{10} \ & =\left(\left(x^{\frac{1}{3}}+1ight)-\left(\frac{\sqrt{x}+1}{\sqrt{x}}ight)ight)^{10} \ & =\left(x^{\frac{1}{3}}-\frac{1}{\sqrt{x}}ight)^{10} \ & T_{r+1}={ }^{10} C_r(x)^{\frac{10-r}{3}}(-1)^r(x)^{-\frac{r}{2}} \ & \frac{10-r}{3}-\frac{r}{2}=0 \ & (20-2 r)-3 r=0 \ & r^{10}=4 \ & \Rightarrow{ }^{10} C_4(-1)^4=210\end{aligned}$

The term independent of $x$ in the expansion of $\left(\frac{(x+1)}{\left(x^{2 / 3}+1-x^{1 /…

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