## Task B-4.2.

How many rational terms are there in the expansion of the binomial $(\sqrt[6]{20}-\sqrt[4]{17})^{2016}$?
## Solution.

The binomial expansion has 2017 terms, all of the form:

$$
\binom{2016}{k}(\sqrt[6]{20})^{2016-k}(-\sqrt[4]{17})^{k}, \quad k \in\{0,1,2, \ldots, 2016\}
$$

For a term to be rational, the numbers $(\sqrt[6]{20})^{2016-k}$ and $(-\sqrt[4]{17})^{k}$ must be rational, which is only possible if $\frac{2016-k}{6}$ and $\frac{k}{4}$ are integers.

We conclude that the number $k$ must be divisible by both 4 and 6, i.e., by 12. 2 points

From 0 to 2016, there are $1+\frac{2016}{12}$ numbers divisible by 12.

Thus, there are a total of 169 rational terms. 2 points