Question

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

Answer 1

General term, $T_{rH}{=}^{15}{C}_r\left(3^{1/5}{)}^r\right(2^{1/3}{)}^{15-r}$

${\Rightarrow }^{15}{C}_r\left(3^{r/5}\right)\left(2^{5-\frac{r}{3}}\right);0\le r\le 15$

For rational terms, $r$ should be a multiple of $3$ and $5$ both

$\Rightarrow r=0,15$

$\Rightarrow$ Sum ${=}^{15}{C}_6{.2}^5{+}^{15}{C}_{15}{.3}^3$

$=32+81=113$