The number of rational terms in the expansion of (2^1/3+5^1/4)^200 is

The general term is $T_{r+1}=\binom{200}{r}\,2^{(200-r)/3}\,5^{r/4}$ and the binomial coefficient is an integer. The term is rational exactly when both exponents are integers, that is when $4\mid r$ and $3\mid(200-r)$. Now $200\equiv2\pmod3$, so we need $r\equiv2\pmod3$ and $r\equiv0\pmod4$. By the Chinese remainder theorem $r\equiv8\pmod{12}$. The admissible values are $r=8,20,32,\dots,200$, an arithmetic progression with common difference $12$, and the last term $200=8+12\cdot16$. So there are $16+1=17$ rational terms.