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.
2026-04-03 08:14:56.1775204096
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.
118 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
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.