Question

For the expansion of $(\sqrt{2}+3^{\frac{1}{5}}{)}^{10},$ which of the following is (are) TRUE?

• A. Sum of rational terms is $41$
• B. There are two rational terms.
• C. There are four rational terms.
• D. Sum of rational terms is $61$

Answer 1

Answer: (A), (B)

There are two rational terms.

General term $(r+1{)}^{th}$ term is
${}^{10}{C}_r\cdot 2^{\frac{10-r}{2}}{3}^{\frac{r}{5}}$
This is rational if $10-r$ is even or zero and $r$ is multiple of $5$ or $r$ is zero
So,
if $r=5$ then $10-r$ is not even
if $r=10$ then $10-r=0$
if $r=0$ then $10-r=10$ an even number
So, this will be rational only if $r=0,10$
So number of rational terms is $2$
Sum of rational terms $={}^{10}{C}_0\cdot 2^5+{}^{10}{C}_{10}\cdot 3^2=41$