Find the number of terms in the
expansion of `(root(2)(9) + root(2)(8) )^(500) ` which are integers .

Explanation

Since , `(root(2)(9) + root(2)(8) )^(500) = (9^(1//4)+ 8^(1//6))^(500) = (3^(1//2) + 2^(1//2))^(500) `
`[ because a , b in ` prime numbers ]
`therefore ` General term , `T_(r+1) = ""^(500)C_(r) (3^(1//2))^(500-r) * (2^(1//2) )^(r)`
`""^(500)C_(r) -3""^( (500-r)/(2) )* 2^(r//2)`
` = ""^(500)C_(r) * 3^(256-r//2) 2^(r//2)`
Now , ` 0 le r le 500 `
For r = 0,2,4,6,8,..,500 indices of 3 and 2 are positive intergers
Hence , number of terms which are inergers = 250+ 1 = 251 .