Question

The number of rational terms in the expansion of ${(9^{1/4}+4^{1/6})}^{1000}$

• A. $167$
• B. $153$
• C. $247$
• D. $183$

Answer 1

The correct option is A $167$

${(9^{1/4}+4^{1/6})}^{1000}={(3^{2/4}+2^{2/6})}^{1000}$
General term
${}^{1000}{C}_r{\left(3^{1/2}\right)}^{1000-r}\times {\left(2^{1/3}\right)}^r$
$={}^{1000}{C}_r\times 3^{500-r/2}\times 2^{r/3}$

When $\frac{r}{2}$ and $\frac{r}{3}$ is an integer then the term will be rational. This means that r is multiple of both $2$ and $3$, that is, $r$ is a multiple of $6$.
Total number of multiples of $6$ between $0$ and $1000$ is $167$.
$\therefore$ Total number of rational terms is $167$.