a > 0, b > 0, x > 0 則6ax^(-7) - 12bx^(-13) = 0   if and only if x^6 = 2b/a   if and only if x = ± (2b/a)^(1/6)?

6ax^(-7) - 12bx^(-13) = 6x^(-13)(ax^6-2b)

因 6x^(-13) > 0, 而實數並無零因子, 也就是說:

若 pq = 0 則 p = 0 或 q = 0.

所以

    6ax^(-7) - 12bx^(-13) = 0

       if and only if  ax^6 - 2b = 0

       if and only if  x^6 = 2b/a

又: x^m = r, 其中 m 是正整數,  則

(1) x = r^(1/m), 或

(2) m 是偶數, x = - r^(1/m).

所以:

    x^6 = 2b/a  if and only if  x = ± (2b/a)^(1/6)