The number of irrational terms in the expansion of
`(root(8)(5)+ root(6)(2))^(100)` is

Explanation

Since `(root(8)(5) + root(6)(2))^(100) = (5^(1//8) + 2^(1//6))^(100)`
` therefore ` General term , ` T_(n+ 1) = ""^(100)C_(r)(5^(1//8) )^(100-r) (2^(1//6))^(r) `
`= ""^(100)C_(r) (5)^((100-r)//8). (2)^(r//6)`
As , 2 and 5 are coprime
` therefore T_(r+1)` will be rational , if (100 - r) is a multiple of 8 and and r is a multiple of 6 .
Also , ` - le r le 100`
` therefore r = 0 , 6,12,18,..,96 `
Now , 100- r = 4 , 10,16,..,100 ...(i)
and 100- r = a , 8, 16, 24 ,.., 100 ...(ii)
The common terms in Eqs (i) and (ii) are 16, 40, 64, and 88
` therefore r = 84,60,36,12 ` given rational terms .
` therefore ` The number of irrational terms = 101- 4 = 97 .