Question

The number of rational terms in the expansion of ${(1+\sqrt{2}+\sqrt[3]{33})}^6$ is

• A. $6$
• B. $7$
• C. $5$
• D. $8$

Answer 1

The correct option is A $6$

The general term of the above expression will be
${}^6{C}_r(\sqrt{2}+3^{\frac{1}{3}}{)}^r$
$={}^6{C}_r{}^r{C}_k{2}^{\frac{r-k}{2}}{3}^{\frac{k}{3}}$
Where
$rϵ[0,6]$ and $k\le r$.
Hence for rational terms, $k$ has to be a multiple of $3$.
Therefore $k=0,3,6$
For
$k=0$, $r$ has to be a multiple of $2$.
Therefore
$k=0;r=0,2,4,6$
For $k=3;r=5$
For $k=6;r=6$
Hence in total we have $6$ rational terms.