Question

Number of irrational terms in the expansion of ${(5^{\frac{1}{6}}+2^{\frac{1}{8}})}^{100}$ is

• A. $96$
• B. $97$
• C. $98$
• D. $99$

Answer 1

The correct option is B $97$

$T_{r+1}={}^{100}{C}_r{5}^{(r-100)/6}{2}^{r/8}$
Hence we get rational terms when

$r=8k$ where k is an integer and

$\frac{8k-100}{6}$ is an integer

$r=16,40,64,88$
Hence we get in total 4 rational terms.
However, total number of terms will be $101$
Hence total number of irrational terms is $101-4$
$=97$ terms.