Let α>0 be the smallest number such that the expansion of x23+2x330 has a term βx−α,β∈ℕ. Then α is equal to _____.

Given x23+2x330Tr+1=30Crx2330−r2x3r30Cr2rx20−11r3 When r=6 we have the smallest α which is 2x20−11(6)3=x−2

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

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answer is 2.

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Detailed Solution

Given ${(x^{\frac{2}{3}} + \frac{2}{x^{3}})}^{30} T_{r + 1} =^{30} C_{r} {(x^{\frac{2}{3}})}^{30 - r} {(\frac{2}{x^{3}})}^{r}$
$30 C_{r} 2^{r} x^{20 - \frac{11 r}{3}}$
When r=6 we have the smallest $α$ which is 2
$x^{\frac{20 - 11 (6)}{3}} = x^{- 2}$