Binomial Expansion

We are asked to find the sum of the rational terms in the binomial expansion of .
The general term in the binomial expansion of is given by: For our expression, , , and , so the general term is: Simplifying:
Step 1: The rational terms in the binomial expansion occur when both exponents of 2 and 3 are integers. Therefore, we need both and to be integers. This implies that:
- must be even, so must be even.
- must also be divisible by 5.
Step 2: The values of that satisfy both conditions are . - For : - For :
Step 3: The sum of the rational terms is the sum of and :