Question

Find the set of '$x$' for which the binomial expansion of ${(2+3x)}^{\frac{2}{3}}$ is valid.

Answer 1

Fact: Binomial expansion of $(x+a{)}^{p/q}$, where $p$ and $q$ are coprime is valid if $\left|\frac{x}{a}\right|<1$

So binomial expansion of ${(2+3x)}^{\frac{2}{3}}$ is valid

If $\left|\frac{3x}{2}\right|<1$

$\Rightarrow |x|<\frac{2}{3}$

$\Rightarrow -\frac{2}{3}<x<\frac{2}{3}$