Prove that the cube root of any irrational number is irratio

Prove that the cube root of any irrational number is irrational

Solution

You can prove it using \"Reductio ad absurdum\".

Let\'s suppose that a is rational.
Then by definition, there exist two positive integers n and p, such as n and p are coprime and:
a = n / p

Let\'s cube that equality:
(a)³ = a = n³ / p³
Since n and p are coprime, so should be n³ and p³.

But the previous equality implies:
n³ = a.p³
which contradicts the fact that n³ and p³ are coprime!!!

Thus the initial assumption that a is rational is false!
Since a is not rational, it must be irrational.