Explanation

[Solution by Dr. Abubacker] The official key $612$ corresponds to the expansion of $\left(1+2^{1/3}+3^{1/2}\right)^6$: the general term is $\frac{6!}{a!\,b!\,c!}\,2^{b/3}\,3^{c/2}$, rational when $3\mid b$ and $2\mid c$. The admissible $(a,b,c)$ are $(6,0,0),(4,0,2),(2,0,4),(0,0,6),(3,3,0),(1,3,2),(0,6,0)$, with terms $1,45,135,27,40,360,4$, i.e. giving $1+45+135+27+40+360+4=612$. (Medium confidence: the exponent printed in the question is $2^{1/2}$, which would give $1296$; the key matches $2^{1/3}$.) Answer: 612.