Explanation

(a)
$Now,\ \left\lbrack \frac{(x + 1)}{x^{2/3} - x^{1/3} + 1} - \frac{(x - 1)}{x - x^{1/2}} \right\rbrack^{10}$
$= \left\lbrack \frac{(x^{1/3})^{3} + 1^{3}}{x^{2/3} - x^{1/3} + 1} - \frac{\left( {(\sqrt{x})}^{2} - 1 \right)}{\sqrt{x}(\sqrt{x} - 1)} \right\rbrack^{10}$
$= \lbrack x^{1/3} + 1 - (x^{- 1/2} + 1\rbrack^{10}$
$= \lbrack x^{1/3} + x^{- 1/2}\rbrack^{10}$
$\therefore T_{r + 1} = \ ^{10}C_{r}(x)^{\frac{10 - r}{3}}( - x^{- 1/2})^{r}$
$= \ ^{10}C_{r}( - 1)^{r}x^{\frac{20 - 5r}{6}}$
For the term independent of $x$
$Put\ \frac{20 - 5r}{6} = 0$
$\Longrightarrow r = 4$
$\therefore\$Required coefficient $= \ ^{10}C_{4} = 210$