What is the number of rational terms in the expansion of (3^...

The general term in the expansion of $\left(3^{\dfrac{1}{2}} + 5^{\dfrac{1}{4}}\right)^{12}$ is given by:

T_{r+1} = ^{12}C_{r} \left(3^{$\dfrac{1}{2}$}\right)^{12-r} \left(5^{\dfrac{1}{4}}\right)^{r}

$T_{r+1} = ^{12}C_{r} 3^{\dfrac{12-r}{2}} 5^{\dfrac{r}{4}}$

For the term to be rational, the exponents of $3 and 5$ must be integers. Therefore, $\dfrac{12-r}{2}$ and $\dfrac{r}{4}$ must be integers.

This requires $r$ to be a multiple of $4$.

Since $0 \le r \le 12$, the possible values of $r$ are $0, 4, 8, 12$.

There are $4$ such values of $r$, which means there are $4$ rational terms in the expansion.

Answer: $4$