The term independent of x in the expansion of

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

Solution:

$\left(\left(x^{1 / 3}+1\right)-\left(\frac{\sqrt{x}+1}{\sqrt{x}}\right)\right)^{10}$

$\left(\mathrm{x}^{1 / 3}-\mathrm{x}^{-1 / 2}\right)^{10}$

$\mathrm{T}_{\mathrm{r}+1}={ }^{10} \mathrm{C}_{\mathrm{r}}\left(\mathrm{x}^{1 / 3}\right)^{10-\mathrm{r}}\left(-\mathrm{x}^{-1 / 2}\right)^{\mathrm{r}}$

$\frac{10-r}{3}-\frac{r}{2}=0 \Rightarrow 20-2 r-3 r=0$

$\Rightarrow \mathrm{r}=4$

$\mathrm{T}_{5}={ }^{10} \mathrm{C}_{4}=\frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1}=210$

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