No positive ℚ = a s.t a*a*a = 2

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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= 2


Does my proof work?
 

Answers and Replies

  • #2
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Something went wrong with formatting of the formulas, especially in the second step, but the overall idea is good.
 
  • #3
Ray Vickson
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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= 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.
 

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