Prove for any $a > 1$, $n\in \mathbb{N}$, $a^\frac{1}{n} >1$
How do i prove this?
Prove for $a > 1$, $n\in \mathbb{N}$, $a^\frac{1}{n} >1$
1
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real-numbers
2 Answers
1
It's probably easier to prove that if $0 Once you have that, you can prove that if $a^{\frac1n}\leq 1$, then $a\leq 1$, which you know because $a=\left(a^\frac1n\right)^n$. This, along with knowing that $A\implies B$ is the same as $\neg B\implies \neg A$, should be all you need.
2
Let $x=a-1>0$ and $n\in \mathbb{N}$, Bernoulli rule shows $$a^\frac{1}{n}=(1+x)^\frac{1}{n}\geq1+\frac{1}{n}x>1$$