Prove that 5^(2/3) is irrational

[P-Illiterate]

Homework Statement

Prove that 5^(2/3) is irrational

Homework Equations

The Attempt at a Solution

I tried writing a proof but that is not getting me any where.

This is what I did so far -

Show that 5^2/3 is irrational

Proof: Suppose that 5^2/3 is rational:
5^2/3 = a/b
^52/33/2 = a^3/2/b^3/2
5(b^3/2) = a^3/2
Substitute a = 2n + 1, b = 2m + 1;

I don't know if I'm suppose to do like this?

 

[Mark44]

Homework Statement

Prove that 5^(2/3) is irrational

Homework Equations

The Attempt at a Solution

I tried writing a proof but that is not getting me any where.

This is what I did so far -

Show that 5^2/3 is irrational

Proof: Suppose that 5^2/3 is rational:
5^2/3 = a/b
5^{2/3 3/2} = a^3/2/b^3/2
5(b^3/2) = a^3/2
Substitute a = 2n + 1, b = 2m + 1;

I don't know if I'm suppose to do like this?

It might be simpler to use the fact that 5^2/3 = $\sqrt[3]{25}$

 

[P-Illiterate]

It might be simpler to use the fact that 5^2/3 = $\sqrt[3]{25}$

Gracias

 

[Mark44]

De nada