Question

In the expansion of ${(7^{\frac{1}{3}}+{11}^{\frac{1}{9}})}^{6561}$

• A. There are exactly $730$ rational terms
• B. There are exactly $5831$ irrational terms
• C. The term which involves greatest binomial coefficients is irrational
• D. The term which involves greatest binomial coefficients is rational

Answer 1

Answer: (A), (C)

The correct options are
A There are exactly $730$ rational terms
C The term which involves greatest binomial coefficients is irrational

General term is ${}^{6561}{C}_r\left(7^{\frac{6561-r}{3}}\right){11}^{\frac{r}{9}}.$
Terms will be rational if $r$ is a multiple of $9$.
$\therefore r=0,9,18,27,....,6561.$
Now $6561=0+(n-1)9\Rightarrow n=730$
Hence, there are $730$ rational terms. The greatest binomial coefficients are ${}^{6561}{C}_{\frac{6561-1}{2}}$ and ${}^{6561}{C}_{\frac{6561+1}{2}}$
$\Rightarrow {}^{6561}{C}_{3280}$ and ${}^{6561}{C}_{3281}$
Now, $3280$ are $3281$ are not a multiple of $3$; hence, both terms involving greatest binomial coefficients are irrational.