Prove that for any natural number $3\leq n$ and rational number t, such that $0<t<1$ , $(t^{n}+1)^{\frac{2}{n}}$ is irrational or provide a counterexample.
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I smell an application of Fermat's Last Marginal Scribble... – Chappers Sep 15 '15 at 23:57
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If $a$ is a rational number such that $a^{n/2}$ is an integer then $a$ is an integer. Proof: If $\sqrt{a^n}$ is an integer then so is $a^n$. Now if $a^n$ is an integer then so is $a$ (given that $a$ is rational). A proof of this can be obtained using the rational root theorem on the polynomial $x^n-(a^n)$ which has only integer solutions since the leading term is $1$.
Therefore if $(t^n+1)^{(2/n)}$ is rational it is an integer.But $t^n+1$ is in the interval $(1,2)$ and the rational values for $\sqrt{a^{n}}$ when $a$ is an integer are all integers.
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How you proof that if $a^{\frac{2}{n}}$ is rational, then a is intigers? – Jamal Farokhi Sep 16 '15 at 00:25