Question:
The sum of all those terms which are rational numbers in the expansion of $\left(2^{1 / 3}+3^{1 / 4}\right)^{12}$ is:
Correct Option: , 4
Solution:
$T_{r+1}={ }^{12} C_{r}\left(2^{1 / 3}\right)^{r} \cdot\left(3^{1 / 4}\right)^{12-r}$
$\mathrm{T}_{s+1}$ will be rational number
when $r=0,3,6,9,12$
$\& \mathrm{r}=0,4,8,12$
$\Rightarrow \mathrm{r}=0,12$
$\mathrm{T}_{1}+\mathrm{T}_{13}=1 \times 3^{3}+1 \times 2^{4} \times 1$
$=24+16=43$