Prove that there is no positive ℚ = a s.t a*a*a= 2
The Attempt at a Solution
If a = j/k a is in lowest form then one of j or k is odd.
(j^3/k^3) = 2 = j^3=2k^3 letting k^3 = z,
j^3 = 2z so j is even because an even number squared is even, thus an even number cubed is even.
Let j = 2i
so 8i^3=2k^3 = 2(2i^3) = k^3
so k is even for same reason as above.
Because k and j are both even, there is no positive ℚ = a s.t a*a*a= 2
Does my proof work?