Let α > 0 , be the smallest number such that the expansion of...

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

${T}_{r + 1} = Cr 30 {\left({x}^{2 / 3}\right)}^{30 - r}{\left(\dfrac{2}{{x}^{3}}\right)}^{r}$

$⇒ {T}_{r + 1} = Cr 30 \cdot {2}^{r} \cdot x\dfrac{60 - 11r}{3}$

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

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

$⇒ 11r > 60$

$⇒ r > \dfrac{60}{11} ⇒ r = 6$

So,

${T}_{7} = C6 30 \cdot {2}^{6}{x}^{- 2}$

We have also observed $\beta = C6 30 {\left(2\right)}^{6}$ is a natural number.

$∴ \alpha = 2$