## Task B-4.3.

How many rational terms are there in the expansion of the binomial $(\sqrt{2}+\sqrt[4]{2})^{2021}$?
## Solution.

It holds that:

$$
(\sqrt{2}+\sqrt[4]{2})^{2021}=\sum_{k=0}^{2021}\binom{2021}{k}(\sqrt{2})^{2021-k} \cdot(\sqrt[4]{2})^{k}
$$

that is, the general term of the binomial expansion is equal to

$$
\binom{2021}{k}(\sqrt{2})^{2021-k} \cdot(\sqrt[4]{2})^{k}=\binom{2021}{k} \cdot 2^{\frac{4042-k}{4}}
$$

The mentioned general term will be rational for $\frac{4042-k}{4} \in \mathbb{N}$, that is, if $4042-k$ is divisible by 4, for $k \in\{0,1, \ldots, 2021\}$.

Since 4042 gives a remainder of 2 when divided by 4, it follows that $k$ must also give a remainder of 2 when divided by 4, so $k \in\{2,6,10, \ldots, 2018\}$.

Since $2=4 \cdot 0+2,6=4 \cdot 1+2, \ldots, 2018=4 \cdot 504+2$, there are 505 such rational terms.