Difference between $\sqrt [q]{x^p}$ and $x^{p/q}$? 
Is there difference between $\sqrt [q]{x^p}$ and $x^{p/q}$ ?

There seems to be no difference when just dealing with positive reals, but what about when $x$ is negative (or even complex)?
p.s. Maybe there's in fact no difference but I'm too suspicious, I just want to be sure...

Some people prefer to consider $\sqrt[n]{\cdot}$ as the inverse of $(\cdot)^n$. In particular, they write freely $\sqrt[3]{-1}$. On the other hand they prefer not to write $(-1)^{\frac{1}{3}}$ because this would be different than $(-1)^{\frac{2}{6}}$.
I have seen this distinction mainly in high-school teachers, because at that level it might be hard to explain that power rules may be false for negative numbers.
Another remark is that, for $x>0$ and $\alpha \in \mathbb{R}$, $x^\alpha = e^{\alpha \log x}$, and this should be true when $\alpha = \frac{p}{q} \in \mathbb{Q}$. There are people who read $x^{\frac{p}{q}}$ as a special case of $x^\alpha$, and they want $x > 0$. It s therefore useful to distinguish between $\sqrt[q]{x}$ and $x^{\frac{1}{q}}$, because the first may be defined also for $x<0$.
