Question

Let $\alpha>0$, be the smallest number such that the expansion of $\left(x^{\frac{2}{3}}+\frac{2}{x^3}\right)^{30}$ has a term $\beta x^{-a}, \beta \in N$.
Then $α$ is equal to _________.

Answer 1

The correct answer is 2.
Tr+1​=30Cr​(x2/3)30−r(x32​)r
=30Cr​⋅2r⋅x360−11r​
360−11r​<0⇒11r>60⇒r>1160​⇒r=6
T7​=30C6​⋅26x−2
We have also observed β=30C6​(2)6 is a natural number.
∴α=2