The sum of the rational terms of 21/5+320is

# The sum of the rational terms of ${\left({2}^{1/5}+\sqrt{3}\right)}^{20}$is

1. A

$71$

2. B

$85$

3. C

$97$

4. D

none of these

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### Solution:

${T}_{r+1,}$the $\left(r+1\right)$th term in the expansion of ${\left({2}^{1/5}+\sqrt{3}\right)}^{20}$is

${T}_{r+1}{=}^{20}{C}_{r}{\left({2}^{1/5}\right)}^{20-r}{\left({3}^{1/2}\right)}^{r}{=}^{20}{C}_{r}{2}^{4-r/5}\left(3{\right)}^{r/2}$
As $2$ and $3$ are relatively prime ${2}^{4-r/5}$ and ${3}^{r/2}$ will be rational if both $4-r/5$ and $r/2$ are integers, i.e. if $r/5$ and $r/2$ are integers. This is possible if
$r=0,10,20.$
Thus, the sum of rational terms is

As this is more than  or $97$, answer is (4).

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