Question

prove that 6^1/3 is irrational​

Answer 1

Answer:

Assume that 6^(1/3) is rational. Then it can be written as 6^(1/3) = n/m for some integers n and m which are co-prime.

So 6 = n³/m³

So n³ must be divisible by 6 and hence n must be divisible by 6. Let n = 6p for some integer p.

This gives:

6= (6p)³/m³

1 = 6²p³/m³

So m³ and hence m must be divisible by 6. But n and m where co-prime so they can not have any factors in common so we have a contradiction. So 6^(1/3) must not be rational. Hence it is irrational.