Animated Solution for Mathematics - Binomial Theorem: Let $\alpha >0$, be the smallest number such that the expansion of $(x^{2/3}+\frac{2}{x^3}{)}^{30}$ has a term $\beta x^{-\alpha },\beta \in \mathbb{N}$. Then $\alpha$ is equal to ______.

We are tasked with analyzing the expression $(x^{2/3}+\frac{2}{x^3}{)}^{30}$ to find a term of the form $\beta x^{-\alpha }$, where $\beta$ is a natural number and $\alpha$ is the smallest positive value possible. This requires a systematic decomposition of the binomial structure.

Every binomial expansion follows a specific rhythm defined by the general term formula:

In this structure, our parameters are $n=30$, $a=x^{2/3}$, and $b=2x^{-3}$. Substituting these into the formula, we establish the foundation:

To simplify the expression, we apply the laws of exponents. First, we process the powers of $x$:

Next, we distribute the power $r$ to the second term:

Combining these components by adding the exponents of $x$, we obtain:

Finding a common denominator of $3$, the exponent simplifies to:

Thus, the general term is expressed as:

The problem defines our target term as $\beta x^{-\alpha }$. By comparing this to our derived expression, we identify the exponent of $x$ as $-\alpha$:

We are given the constraint that $\alpha$ must be the smallest positive number. Therefore, we set the condition $\alpha >0$:

Solving for $r$:

Since $r$ must be an integer, the smallest integer satisfying $r>5.45$ is $r=6$. We now calculate the value of $\alpha$ using this integer:

By navigating the constraints and the algebra, we have determined that the smallest positive value for $\alpha$ is $2$.