Prove that following numbers are not rational :

(6)^1/3

Suppose 6^1/3 is rational.

Then, 6^1/3 = for some integers n and m which are co-prime.

So, 6 =

6m³ = n³

So, n³ must be divisible by 6

n must be divisible by 6.

Let n = 6p for some integer p

This gives

6 =

1 =

m³ is divisible by 6

Hence, m must be divisible by 6.

But n and m where co-prime.

So, we have a contradiction.

Hence, (6)^1/3 is irrational