Question

The number of rational terms in the expansion of ${(\sqrt{3}+\sqrt[4]{45})}^{124}$ is

• A. $31$
• B. $32$
• C. $33$
• D. $34$

Answer 1

Answer: (B)

$32$

$T_r={}^{124}{C}_{r-1}\left(\sqrt{3}{)}^{125-r}\right(\sqrt[4]{45}{)}^{r-1}$
When both the terms are rational , $T_r$ will be rational.
Hence, $\frac{125-r}{2}$ and $\frac{r-1}{4}$ both must be integers.
Therefore, $r$ must be of the form $4k+1$, where $k$ is an integer.
There are $125$ terms in the expansion. Hence, $k$ can assume values from $0$ to $31.$ Hence, there are $32$ values of $k$ and $32$ rational terms in the expansion.
Hence, option $B$ is correct.