The sum of all rational terms in the expansion of $\left(1 + 2^{1/2} + 3^{1/2}\right)^6$ is equal to
Step-by-Step Solution
Key Concept: Use multinomial expansion; terms are rational when powers of $2^{1/2}$ are even (i.e., $\beta$ divisible by 2) and powers of $3^{1/2}$ are even (i.e., $\gamma$ divisible by 3).
General term is $\frac{6!}{\alpha!\beta!\gamma!}(1)^\alpha(2^{1/2})^\beta(3^{1/2})^\gamma$. For rational terms: $3|\beta$ and $2|\gamma$. Valid $(\beta, \gamma, \alpha)$: $(0,0,6)\to 1$, $(0,2,4)\to 45$, $(0,4,2)\to 135$, $(0,6,0)\to 27$, $(3,0,3)\to 40$, $(3,2,1)\to 360$, $(6,0,0)\to 4$. Sum $= 1+45+135+27+40+360+4 = 612$.
Correct Answer: 612