Let x 1 , … , x n be positive rational numbers. If x 1 l 1 , … , x n l n are all irrational numbers (where l 1 , l 2 , … , l n ∈ N ∗ ), does it follow that x 1 l 1 + ⋯ + x n l n is an irrational number, too?

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Let       x    1    ,  …  ,      x    n   be positive rational numbers. If             x      1              l      1        ,  …  ,            x      n              l      n       are all irrational numbers (where       l    1    ,      l    2    ,  …  ,      l    n    ∈            N        ∗  ), does it follow that            x      1              l      1        +  ⋯  +            x      n              l      n      is an irrational number, too?

iletsa2ym · Accepted Answer

Problem: If       a    1    ,  .  .  ,      a    k   are positive integers, and   m is a positive integer, so that             a      i        m    ∉      Q   then  ∑            a      i        m    ∉      Q    You can do the reduction to this problem in two steps:Step   1: Let   m  =  l  c  m  (      l    1    ,  .  .  ,      l    n    ). Then       1          l      i        =            k      i        m  . Let       y    i    =      x    i                  k        i            .Then, you know that       y    i   are positive rational numbers and       x    i          1              l        i              =            y      i        m  .Step   2: Let       y    i    =            b      i              c      i       and let   d  =  l  c  m  (      c    1    ,  .  .  ,      c    n    ). Then you can write      y    i    =            a      i              d      m

Let x1,…,xn be positive rational numbers. If root(l_1)(x_1),…,root(l_n)(x_n) are all irrational numbers (where l_1,l_2,…,l_n in N∗), does it follow that root(l_1)(x_1),…,root(l_n)(x_n) is an irrational number, too?

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