Question

Find the sum of all rational terms in the expansion of ${(3^{1/5}+2^{1/3})}^{15}$.

Answer 1

$T_{r+1}={}^{15}{C}_r{3}^{3-\frac{r}{5}}{2}^{\frac{r}{3}}$
Hence only $T_1$ and $T_{16}$ will be the rational terms.
$T_{0+1}={}^{15}{C}_0{3}^3$
$=27$ $...\left(a\right)$
$T_{15+1}={}^{15}{C}_{15}{2}^5$
$=32$ $...\left(b\right)$
Hence, $a+b$ implies
$27+32$
$=59$