Question

Find the number of rational terms in the expansion of ${\left(7^{1/3}+11^{1/9}\right)}^{6561}$.

Answer 1

Answer: 730.00

$T_{r+1}={}^{6561}{C}_r{\left(7^{1/3}\right)}^{6561-r}\cdot {\left(11^{1/9}\right)}^r$
$={}^{6561}{C}_r{7}^{2187}\cdot 7^{-r/3}\cdot 11^{r/9},0\le r\le 6561$
Now $T_r+1$ will be rational when $r/3$ and $r/9$ both are integers.
This is possible only when $r$ is a multiple of $9$.
But multiple of $9$ from $0$ to $6561$ are
$0,9,18,27,\dots \dots \dots ,6561$
These are in $AP$, so their number $n$ is given by
$6561=0+(n-1)9$
$\Rightarrow n=730$