The sum of rational terms in `(sqrt(2) + sqrt(3) + root (6)(5))^(10)` .is

Explanation

We have , ` (sqrt(2) + root(3)(3) + root(6)(5) )^(10)= (2^(1//2) + 3^(1//3) + 5^(1//6))^(10)`
` = underset(alpha + beta + gamma = 10)sum(10!)/(alpha!beta!gamma!) 2^(alpha//2)*3^(beta//3) *5^(gamma//6)`
For rational terms
` alpha = 0,2,4,6,8,10, beta = 1,3,6,9, gamma = 0,6 `
Since , ` 0 le alpha , beta ,gamma le 10`.
` therefore ` Possible triplets are (4,0,6),(4,6,0),(10,0,0).
There exists three rational terms .
`= (10!)/(4!0!6!) 2^(2) *5 + (10!)/(4!6!0!) 2^(2) *3 + ^(2)(10!)/(10!0!0!) 2^(5)`
` = 4200 + 7560 + 32 = 11792 ` .