Question

In the expansion of $(\sqrt{2}+\sqrt[3]{35}{)}^{20}$ the number of rational terms will be:

• A. $3$
• B. $10$
• C. $4$
• D. $8$

Answer 1

The correct option is C $4$

Given

$(\sqrt{2}+\sqrt[3]{35}{)}^{20}$

General term of given expamsion

$T_{r+1}={}^{20}{C}_r\left(\sqrt{2}{)}^{20-r}\right(\sqrt[3]{35}{)}^r$

$T_{r+1}={}^{20}{C}_r\left(2{)}^{\frac{20-r}{2}}\right(5{)}^{\frac{r}{3}}$

Now, $T_{r+1}$ be rational if $\frac{20-r}{2}$ and $\frac{r}{3}$ are integers

Hence $20-r$ Should be multiple of $2$ and $r$ should be multiple of $3$ where $0\le r\le 20$

Hence possible value of $r=0,3,6,9,12,15,18$

$20-r=20,17,14,11,8,5,2$

$20-r$ should be multiple of $2$

Hence $20-r=20,14,8,2$

Hence possible rational terms are $4$