Question

The number of irrational terms in the expansion of $(\sqrt[8]{85}+\sqrt[6]{62}{)}^{100}$ , is:

• A. $100$
• B. $98$
• C. $96$
• D. $97$

Answer 1

The correct option is D $97$

The general term in the expansion ${(5^{1/8}+2^{1/6})}^{100}$ is $T_{r+1}{=}^{100}{C}_r.5^{\frac{100-r}{8}}{2}^{\frac{r}{6}}$

Now, we need to calculate rational terms.

We find HCF of $\frac{100-r}{8}$ and $\frac{r}{6}$

So, $r$ is divisible by $4$ and gives remainder $4$ with $8$.

HCF of 4 and 6 is 12.

Now, check for multiples of 12.

$12$ $36$ $60$ $84$ satisfy the conditions.

So, rational terms $=4$

Total terms$=101$

Irrational terms$=101-4=97$