Expansion frenzy 2

The binomial expansion is $(\sqrt 3 + \sqrt[4] 5)^{200} = \sum_{k=0}^{200} \binom {200} k (\sqrt 3)^{200-k} (\sqrt[4] 5)^k.$ There are 201 terms; however, they are only rational if $k$ is a multiple of 4 and $200-k$ is a multiple of 2. If the former is true, then the latter is also true; thus the rational terms are those with $k = 0, 4, 8, \dots, 200$ .

There are $\boxed{51}$ of these terms.

They are of the form $t_{4n} = \binom{200}{4n}\cdot 9^{50-n}\cdot 5^n,\ \ \ n = 0, 1, 2, \dots, 50.$