(A) `because (sqrt2+3^(1//5))^(10) = (2^(1//2 ) + 3 ^(1//5)) ^(10`
`therefore T_(r+1) = ""^(10)C_(r) cdot 2 ^((10-r)/2) cdot 3 ^(r/5)`
For rational terms, r = 0, 10 `[because 0 le r le 10]`
`therefore` Number of rational terms = 2
i.e., m = 2 and `(sqrt(3)+5^(1//8))^(256) = (3^(1//2) + 5^(1//8))^(256)`
`therefore T_(R+10)= ""^(256)C_(R) cdot 3^((256-R)/2 ) cdot 5 ^(R//8)`
For rational terms, r = 0, 8, 16, 24, 32, ..., `[ because 0 le r le 256]`
`therefore` Number of rational terms = 1 + 32 = 33
i.e., `n = 33 rArr m + n = 35 (s) and n - m = 31`
(B) `T_(r+1) "in" (2^(1//3) + 3^(1//5) )^(40) = ""^(40)C_(r)cdot 2 ^((40-r)/3) cdot 3 ^(r//5)`
For rational terms, r = 10, 25, 40 `[because 0 le r le 40]`
`because` Number of rational terms = 3
` therefore` Number of irrational terms
= Total terms - Number of rational terms
= 41 - 3 = 38 i.e. m = 38
and `T_(R-1) "in" (5^(1//10)+2^(1//6))^(100 ) = ""^(100)C_(R) cdot 5^((100 - R)/10) cdot2 ^(R//6)`
rational terms, R =0, 30 , 60, 90 `[because 0 le R le 100] `
`because` Number of rational terms = 4
`therefore` Number of irrational terms = 101 - 4 = 97
i.e `n = 97 rArr m + n = 100 , n - m = 97 - 38 = 39 `
(C) `because (1 + sqrt(2)+ 3 ^(1//3))^(6) = (1+ 2^(1//2) + 3 ^(1//3))^(6)`
`= sum _(alpha + beta+ gamma= 6) (6!)/(alpha! beta! gamma!) (1) ^(alpha )(2^(1//2))^(beta) (3^(1//3))^(gamma)`
`= sum _(alpha + beta+ gamma= 6) (6!)/(alpha! beta! gamma!) 2^(beta//2) cdot 3 ^(gamma//3)`
Values of `(alpha , beta, gamma)` for rational terms are (0, 0, 6)
(1, 2, 3), (3, 0, 3), (0, 6, 0), (2, 4, 0), (4, 2, 0), (6, 0, 0).
`therefore` Number of rational terms = 7 i.e., m = 7
and `(1+root(3)(2)+ root(5)(3) )^(15) = (1 + 2^(1//3) + 3 ^(1//5))^(15)`
`= sum _(alpha + beta + gamma = 15 ) (15!)/(alpha! beta! gamma!) (1)^(alpha )(2^(1//3))^(beta) (3^(1//5))^(gamma)`
`= sum _(alpha + beta + gamma = 15 ) (15!)/(alpha! beta! gamma!) 2^(beta//3) cdot 3^(gamma//5) `
of `(alpha, beta, gamma)` for rational terms are
95, 0, 10), (2, 3, 10), (10, 0, 5), (7, 3, 5), (4, 6, 5), (1, 9, 5),
(15, 0, 0), (12, 3, 0), (9, 6, 0), (6, 9, 0), (3, 12, 0), (15, 0, 0).
`therefore` Number of rational termas = 13 i.e. n = 13
Hence, `m + n = 20 and n - m = 6`
