Question on double inequality with radicals

Question

A really simple question, but I thought I'd ask anyway. Does $n<x^n<(n+1)$ imply $\sqrt[n] n < x < \sqrt[n] {n+1}$?

Thank you very much.

Answer 1

Not in general, but yes if $x\geq 0$.

Otherwise, take $$ 2 < x^2 < 3 $$ with $x = \frac{3}{2}$.

Answer 2

Note that square roots, 4th roots, 6th roots etc. can turn numbers from positive to negative; if it is so, then NO. But if the root is uniquely defined as mapping positive numbers to positive, then YES.