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

The first term is \(\frac{x+1}{{x}^{2\mathrm{/}3}−{x}^{1\mathrm{/}3}+1}\).

Substituting this into the first term: \(\frac{({x}^{1\mathrm{/}3}+1)({x}^{2\mathrm{/}3}−{x}^{1\mathrm{/}3}+1)}{{x}^{2\mathrm{/}3}−{x}^{1\mathrm{/}3}+1}={x}^{1\mathrm{/}3}+1\)

The second term is \(\frac{x−1}{x−{x}^{1\mathrm{/}2}}\).

Substituting these: \(\frac{(\sqrt{x}−1)(\sqrt{x}+1)}{\sqrt{x}(\sqrt{x}−1)}=\frac{\sqrt{x}+1}{\sqrt{x}}\)\(=\frac{{x}^{1\mathrm{/}2}+1}{{x}^{1\mathrm{/}2}}=1+{x}^{−1\mathrm{/}2}\)

\({(({x}^{1\mathrm{/}3}+1)−(1+{x}^{−1\mathrm{/}2}))}^{10}\)\(={({x}^{1\mathrm{/}3}−{x}^{−1\mathrm{/}2})}^{10}\)

\({T}_{r+1}=(\frac{10}{r})({x}^{1\mathrm{/}3}{)}^{10−r}(−{x}^{−1\mathrm{/}2}{)}^{r}\\ =(\frac{10}{r})(−1{)}^{r}{x}^{\frac{10−r}{3}−\frac{r}{2}}\)

For the term to be independent of \(x\), the exponent of \(x\) must be zero:

\(\frac{10−r}{3}−\frac{r}{2}=0\) :

\(2(10−r)−3r=0\)

\(20−2r−3r=0\)

\(20=5r\text{  }⟹\text{  }r=4\)

Substitute \(r=4\) into the expression for

\({T}_{r+1}\): \({T}_{5}=(\frac{10}{4})(−1{)}^{4}=\frac{10\times 9\times 8\times 7}{4\times 3\times 2\times 1}\times 1=10\times 3\times 7=210\)