The sum of all rational terms in the expansion of (2^(1)/(5)+5^(1)/(3))^15 is equal to :

Question

The sum of all rational terms in the expansion of ${({2}^{\frac{1}{5}}+{5}^{\frac{1}{3}})}^{15}$ is equal to :

Accepted Answer

$$\begin{aligned}
& T_{r+1}={ }^{15} \mathrm{C}_r\left(2^{1 / 5}\right)^{15-r}\left(5^{1 / 3}\right)^r \
& ={ }^{15} C_r 5^{r / 3} 2^{\left(3-\frac{r}{5}\right)}
\end{aligned}$$

For rational terms,

$$\frac{r}{3}$$ and $$\frac{r}{5}$$ must be integer

3 and 5 divide $$r \Rightarrow 15$$ divides $$r \Rightarrow r=0$$ and $r=15$

$${ }^{15} C_0 5^0 2^3+{ }^{15} C_{15} 5^5 2^{(0)}$$

$$\begin{aligned}
& =8+3125 \
& =3133
\end{aligned}$$

Solutions · 1