Question

In the expansion of $\left(5^{\frac{1}{6}}+2^{\frac{1}{8}}\right)^{100}$, which of the following hold(s) good?

• A. The number of irrational terms are 97 .
• B. The number of rational terms are 5.
• C. The binomial coefficient of the term in which exponent of ' 5 ' is 15 is ${ }^{100} C _{90}$.
• D. The term in which the exponent of 2 and 5 will be equal is the $58^{\text {th }}$ term from the beginning.

Answer 1

Answer: (A), (C)

$ T_{r+1}={ }^{100} C_r 5^{\frac{100-}{6}} 2^{\frac{r}{8}}$
$\text { For } \frac{r}{8} \in Q $
$r=0,8,16,24, \ldots \ldots \ldots \ldots . .96$
For $\frac{100-r}{6}$ to be rational
$r =16,40,64,88$
4 terms are rational
$\frac{100-r}{6}=15$
(C) $r =10 $