The sum of all rational terms in the expansion of $\left(1 + 2^{1/2} + 3^{1/2}\right)^6$ is equal to

Step 1: Identify the general term

We begin by writing the general term of the multinomial expansion $(a+b+c)^n$. For the given expression $(1 + 2^{1/2} + 3^{1/2})^6$, the general term is given by the formula shown, where $r_1 + r_2 + r_3 = 6$ and $r_1, r_2, r_3$ are non-negative integers.

Step 2: Simplify the general term for rationality

For a term to be rational, the powers of 2 and 3 must be integers. This means that $r_2/2$ and $r_3/2$ must be integers. Therefore, $r_2$ and $r_3$ must both be even integers.

Step 3: List possible values for $r_2$ and $r_3$

Since $r_2$ and $r_3$ must be even and their sum with $r_1$ cannot exceed 6, the possible even values for $r_2$ and $r_3$ are 0, 2, 4, and 6.

Step 4: Calculate terms for valid $(r_1, r_2, r_3)$ combinations

We systematically list all combinations of even $r_2$ and $r_3$ such that $r_1 = 6 - r_2 - r_3$ is non-negative. For each valid combination, we calculate the corresponding term using the simplified general term formula.

Step 5: Sum all rational terms

Finally, we sum all the calculated rational terms to find the total sum of all rational terms in the expansion.

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