Find the number of irrational terms in the expansion of `(5^(1//6)+2^(1//8))^(100)dot`

Answer

`97`

Explanation

`T_(r+1)`, the `(r+1)` th term in the expansion of `(5^(1//6) + 2^(1//8))^(100)`, is
`T_(r+1) = .^(100)C_(r)(5^(1//6)).^(100-r)(2^(1//8))^(r)`
As 5 and 2 are respectively prime, `T_(r+10` will be rational if `(100-r)//6` and `r//8` are both integers, i.e, `100 -r` is a multiple of 6 and ris a multiple of 8.
As `0 le r le 100`, multiples of 8 up to 100 and corresponding values of 100 - r aer given by
`r = 0, 8, 16, 24,"......",88,96`
`100 -r = 100, 92,84,76,"......",12,4`
In `100 - r` values, multiples of 6 ar `84, 60, 36` and `12`.
Hence, there are just four rational terms. Therefore, the number of irrational terms is `101 - 4 = 97`.