We suppose that $a\in\Bbb R^{+}_0$ we know that
$$a^{\frac{m}{n}}=\sqrt[n]{a^m} \tag 1$$
Is $(1)$ provable or is it a given definition. Many years ago I remember that perhaps there was a proof of such an equality. Does anyone remember it? Thank you all.
I would say that this is a definition for the left hand side. What you are thinking of is probably this line of reasoning.
$(a^{\frac{m}{n}})^n = a^m\iff a^{\frac{m}{n}}=\sqrt[n]{a^m} $.
This shows that if we want the exponential identity $a^{bc}=(a^b)^c$ to hold for rational numbers as well as integers, we need to use this definition for $a^{\frac{m}{n}}$.