Question

The number of irrational terms in the expansion of ${(3^{\frac{1}{8}}+5^{\frac{1}{4}})}^{84}$ is

• A. $73$
• B. $74$
• C. $75$
• D. $76$

Answer 1

The correct option is B $74$

For ${(3^{\frac{1}{8}}+5^{\frac{1}{4}})}^{84}$

$T_{r+1}={}^{84}{C}_r\left(3^{\frac{84-r}{4}}\right)\left(5^{\frac{r}{8}}\right)$

In order to get rational terms $5$ and $3$ both should have integral power.
We know $r$ can take up values from $0$ to $84.$

For $\frac{r}{8}$ and $\frac{84-r}{4}$ to be integer, $r=0,8,16,...,80=11\text{terms}$

The number of rational terms $=11$
Total terms $=85$
$\Rightarrow$ Number of irrational terms $=85-11=74$