Question

The rational term(s) in the expansion of ${(\sqrt{2}+(3{)}^{\frac{1}{5}})}^{10}$ is/are

• A. $3$
• B. $9$
• C. $6$
• D. $32$

Answer 1

Answer: (B), (D)

$32$

$T_{r+1}={}^{10}{C}_r\left(2{)}^{\left(\frac{10-r}{2}\right)}\right(3{)}^{\frac{r}{5}}$
This is rational, if $\frac{r}{5}$ and $(5-\frac{r}{2})$ are integers,
Now, $r$ must be a multiple of $10$
(LCM of $2$ and $5$).
$\therefore r=0,10$
So the rational terms are
${}^{10}{C}_0\cdot 2^5$ and ${}^{10}{C}_{10}\cdot 3^2$
or $32$ and $9$