1. Find the term that does not contain $x$ in the expansion (according to the binomial formula) of $\left(2 x+\frac{\sqrt[3]{x^{2}}}{x}\right)^{n}$ if the sum of all binomial coefficients in this expansion is 256.
Solution. The sum of all binomial coefficients is

$$
\binom{n}{0}+\binom{n}{1}+\ldots+\binom{n}{n}=(1+1)^{n}=2^{n}
$$

so $2^{n}=256=2^{8}$. Therefore, $n=8$. Furthermore, the $k+1$-th term in the expansion is

$$
\binom{8}{k}(2 x)^{8-k}\left(x^{\frac{2}{3}-1}\right)^{k}=\binom{8}{k} 2^{8-k} x^{8-k-\frac{k}{3}}
$$

so it does not contain $x$ if and only if $8-k-\frac{k}{3}=0$. From this, we get that $k=6$. Thus, the desired term is the seventh and is equal to $\binom{8}{6} 2^{8-2}=112$.