Find the number of irrational terms in the binomial expansion of $(3^{1/5}+7^{1/3})^{100}$

Question

After expanding the above term binomially, I can well guess that the majority terms are irrational, but i'm unable to find any proper method of solving this sum, after repeated trials.

Please help

Thank You

Answer 1

Hint: The binomial sum is $$\sum_{k = 0}^{100} {100 \choose k} 3^{k/5}7^{(100-k)/3},$$ and the rational terms are exactly the integer terms, i.e., when both $k/5$ and $(100-k)/3$ are integers.