# No positive ℚ = a s.t a*a*a = 2

## Homework Statement

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?

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

Ray Vickson
Homework Helper
Dearly Missed

## Homework Statement

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?

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.