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

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

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Hint: $\sqrt 2 = 2^{1/2} = (2^{1/12})^6$. If $x := 2^{1/12}$ were rational, what could we say about $x^6$? – martini Jun 24 '12 at 17:38
No you can't assume it, but you can show it. There is a big difference. – Thomas Andrews Jun 24 '12 at 18:23

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