Generalize further and show that for any whole number \(r\), the \(r\) th root \(\sqrt[r]{A}\) is irrational unless \(A\) is an \(r\) th power. Let \(x=\sqrt[r]{A}\). Show that if \(x\) is rational, then we may choose a smallest whole number \(n\) such that \(n x^{j}\) is a whole number for \(j=1, \ldots, r-1\). Then consider \(m=n x-n[x]\) as before.