Is it assumable that $2^{1/12}$ is irrational because $2^{1/2}$ is?

Question

I need to prove that $2^{1/12}$ is irrational but I need to connect this to $2^{1/2}$ being irrational.
I know how to prove that $2^{1/2}$ is irrational, but can I assume that $2^{1/12}$ is irrational because $2^{1/2}$ is and why?

Cheers

Answer 1

If $2^{1/12}$ were rational, then $(2^{1/12})^6$ would also be rational.