TITLE: sum of rational terms in $\left(\sqrt{2}+\sqrt{27}+\sqrt{180}\right)^{10}$
QUESTION [9 upvotes]: Find a sum of the rational terms in the following expression after full expanding. $$\left(\sqrt{2}+\sqrt{27}+\sqrt{180}\right)^{10}$$

Since the term should be rational, each power should be even.
Number of non negative even integers $m,n,q$ satisfying 
$$m+n+q=10$$ gives number of triplets as $21$..but its very tedious to find sum of $21$ terms...any other approach?

REPLY [2 votes]: Let $a=\sqrt2$, $b=3\sqrt3$ and $c=6\sqrt5$.
Hence, the needed number is
$$\sum_{cyc}\left(a^{10}+\binom{10}{8}(a^8b^2+a^8c^2)+\binom{10}{6}(a^6b^4+a^6c^4)+\binom{10}{6,2,2}a^6b^2c^2+\binom{10}{4,4,2}a^4b^4c^2\right)=$$
$$=188971148939+45\cdot30539765574+210\cdot4912928964+$$
$$+1260\cdot322052760+3150\cdot51263280=3162241491809.$$