No positive ℚ = a s.t a*a*a = 2

[r0bHadz]

Homework Statement

Prove that there is no positive ℚ = a s.t a*a*a= 2

Homework Equations

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= 2Does my proof work?

 

[mfb]

Something went wrong with formatting of the formulas, especially in the second step, but the overall idea is good.

 

[Ray Vickson]

Homework Statement

Prove that there is no positive ℚ = a s.t a*a*a= 2

Homework Equations

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= 2Does my proof work?

Your first equation is horribly wrong, just because your were trying to pack too much in a single equation. What you wrote is, essentially, $A/B=C=A=BC$, which is wrong except when $C=1$ and $A = B$. What I hope you meant was "$A/B=C$, hence $A = BC$". Alternatively, you could have written "$A/B=C \Rightarrow A = BC.$" Please tell me you see the difference.