Question

The number of integral terms in the expansion of $(\sqrt{3}+\sqrt[8]{85}{)}^{256}$ is

• A. $32$
• B. $33$
• C. $34$
• D. $35$

Answer 1

The correct option is A $32$

In the expansion of $(\sqrt{3}+\sqrt[8]{85}{)}^{256}$,

General term, $T_{r+1}{=}^{256}{C}_r\left(\sqrt{3}{)}^{256-r}\right(\sqrt[8]{85}{)}^r$

$={}^{256}{C}_r\cdot 3^{(256-r)/2}\cdot 5^{r/8}$

The terms are integral if $\frac{256-r}{2}$ and $\frac{r}{8}$ are both positive integers.

$\therefore r=0,8,16,24,....,256$

Hence, there are $33$ integral terns.