Question

The number of irrational terms in the expansion of $\left( \sqrt[8]{5} + \sqrt[6]{2} \right)^{100}$ is

• A. 97
• B. 98
• C. 96
• D. 99

Answer 1

Answer: (A)

97

Answer 2

$T_{r + 1} =^{100} ⥂ C_{r}5^{\frac{100 - r}{8}}.2^{\frac{r}{6}}$

As 2 and 5 are co-prime. $T_{r + 1}$ will be rational if $100 - r$ is multiple of 8 and r is multiple of 6 also $0 \leq r \leq 100$

$\therefore r = 0,6,12.......96$; $\therefore 100 - r = 4,10,16.....100$ ......(i)

But $100 - r$ is to be multiple of 8.

So, $100 - r$ = 0, 8, 16, 24,......96 .....(ii)

Common terms in (i) and (ii) are 16, 40, 64, 88.

$\therefore$ r = 84, 60, 36, 12 give rational terms

$\therefore$ The number of irrational terms = 101 – 4 = 97.