Question

If $n$ is the number of irrational terms in the expansion of ${(3^{\frac{1}{4}}+5^{\frac{1}{8}})}^{60}$, then $(n-1)$ is divisible by:

• A. $8$
• B. $26$
• C. $7$
• D. $30$

Answer 1

The correct option is B $26$

$T_{r+1}{=}^{60}{C}_r\left(3^{1/4}{)}^{60-r}\right(5^{1/8}{)}^r$
rational if $\frac{60-r}{4},\frac{r}{8}$ both are whole numbers, $r\in \{0,1,2,....60\}$
$\frac{60-r}{4}\in W\Rightarrow r\in \{0,4,8,...60\}\text{and}\frac{r}{8}\in W\Rightarrow r\in \{0,8,16,...56\}$
$\therefore$ Common terms $r\in \{0,8,16,....56\}$
So $8$ terms are rational
Then irrational terms $=61-8=53=n$
$\therefore n-1=52=13\times 2^2$
Factors $1,2,4,13,26,52$