# Provide a proof using the rational roots theorem

Prove that if $c$ is a positive rational and $k$ is a positive integer, then $c^{1/k}$ is either an integer or irrational, using the rational roots theorem.

• it is enough to prove for natural $c$ Oct 15, 2013 at 19:59
• I suppose Matina Manos you mean $c$ is a positive integer!
– P..
Dec 24, 2013 at 11:12

The way you have posed the problem is: Given $c>0$ over rationals, then the $k$-th root of $c$ is irrational (or integer). That's simply not true! For example, if $c=4/9$ and $k=2$, then $\sqrt{4/9}=2/3$ is neither irrational nor an integer.

Well, by definition $c^{1/k}$ is a root of the polynomial $x^k - c$...

• I do not know what that has to do with my question Oct 16, 2013 at 0:50