In the expansion of `(5^(1//2)+7^(1//8))^(1024),` the number of integral terms is `128` b. `129` c. `130` d. `131`

Options

A. 128
B. 129
C. 130
D. 131

Answer

B

Explanation

The general term `T_(r + 1)` in the expansion of
`(5^(1//2) + 7^(1//8)) ^(1024)` is given by
`T_(r+ 1) = ""^(1024)C_(r) (5^(1//2))^(1024 - r)(7 ^(1//8))^(r)`
`rArr T_(r + 1) = ""^(1024)C_(r) 5 ^(512 - r//2) 7^(r//8)`
`rArr T_(r +1) = (1024C_(r) 5^(512 - r)) (5 ^(r//2) 7^(r//8))`
`rArr T_(r+ 1) = { ""^(1024)C_(r) 5^(512 - r)} (5^(4) xx 7 )^(7)^(r //8)`
Clearly, `T_(r +1)` will be an integer , iff
` r//8` is an integer such that 0 `le r le 1024`
`rArr r` is a multiple of 8 lying satisfying ` 0 le r le 1024`
`rArr ` r = 0 , 8 , 16 ..., 1024
`rArr ` r can assume 129 values
Hence, these are 129 integral terms in the expansions of
`(5^(1//2) + 7 ^(1//8))^(1024)`