Can always roots be transformed into exponents?

Question

The problem I find when assuming that for all $x\in \mathbb{R}$ it complies that $\sqrt[n]{x^m}=x^\frac{m}{n}$. But is it always true? Wouldn't this allow me to do this?: $x^\frac{m}{n}=x^{m\cdot \frac{1}{n}}=x^{\frac{1}{n}\cdot m}=(x^\frac{1}{n})^m=(\sqrt[n]{x})^m$. But this shouldn't be always true when working with real numbers, since if $x\in \mathbb{R}$ this would mean that $\sqrt{(x)^3}=(\sqrt{x})^3$, but since $\sqrt{x^3}=\sqrt{x^2 x}=\sqrt{x^2}\sqrt{x}=|x|\sqrt{x}$, and $(\sqrt{x})^3=\sqrt{x}\cdot \sqrt{x} \cdot \sqrt{x}=x\sqrt{x}$, but not always $|x|=x$... does any one has some notes or book chapters where all the roots and powers properties and theorems are proven? Not even for complex numbers some of these properties hold.