The sum of the coefficient of \({x}^{2/3}\) and \({x}^{-2/5}\) in the binomial expansion of \({\left({x}^{2/3}+\frac{1}{…

General term, \({T}_{r+1}={}^{9}C_{r}{({x}^{2/3})}^{9-r}{\left(\frac{{x}^{-2/5}}{2}\right)}^{r}\)

\(={}^{9}\mathrm{C}_{\mathrm{r}}{\left(\frac{1}{2}\right)}^{\mathrm{r}}{\left(\mathrm{r}\right)}^{\left(6-\frac{2\mathrm{r}}{3}-\frac{2\mathrm{r}}{5}\right)}\)

for coefficient of \({x}^{2/3}\),

put \(6-\frac{2r}{3}-\frac{2r}{5}=\frac{2}{3}\)

\(\Rightarrow r = 5\)

Coefficient of \({x}^{2/3}\) \(={}^{9}C_{5}{\left(\frac{1}{5}\right)}^{5}\)

For coefficient of \({x}^{-2/5}\)

put \(6-\frac{2r}{3}-\frac{2r}{5}=-\frac{2}{3}\)

\(\Rightarrow r = 6\)

Coefficient of \({x}^{-2/5}\) is \({}^{9}C_{6}{\left(\frac{1}{2}\right)}^{6}\)

sum \(={}^{9}C_{5}{\left(\frac{1}{2}\right)}^{5}+{}^{9}C_{6}{\left(\frac{1}{2}\right)}^{6}=\frac{21}{4}\)