Let α > 0 , be the smallest number such that the expansion of x 2 3 + 2 x 3 30 has a term β x…

General term of the binomial expansion of ${\left(x^{\frac{2}{3}}+\frac{2}{x^3}\right)}^{30}$ is

$T_{r+1}={}^{30}C_r{\left(x^{2/3}\right)}^{30-r}{\left(\frac{2}{x^3}\right)}^r$

$\Rightarrow T_{r+1}={}^{30}C_r·2^r·x^{\frac{60-11r}{3}}$

So, for term of the form $\beta x^{-\alpha }$, we have

$\frac{60-11r}{3}<0$

$\Rightarrow 11r>60$

$\Rightarrow r>\frac{60}{11}\Rightarrow r=6$

So,

$T_7={}^{30}C_6·2^6{x}^{-2}$

We have also observed $\beta ={}^{30}C_6{\left(2\right)}^6$ is a natural number.

$\therefore \alpha =2$