<s>[INST] Is there a difference between $(x)^{\frac{1}{n}} $ and $\sqrt[n]{x}$? Is there a difference between $(x)^{\frac{1}{n}}$ and $ \sqrt[n]{x}$ ?
I'm confused with this topic. Any ideas or examples ?
If  $(x)^{\frac{1}{n}} = \sqrt[n]{x}$
Consider $x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$ . Is it the same if I write $ x=\frac{-b \pm(b^2-4ac)^{\frac{1}{2}}}{2a}$ ?
 [/INST] There is no difference.  In the same way as I write $e^x$ when $x$ is simple enough, and $\exp(x)$ otherwise, I also write $\sqrt{x}$ when $x$ is simple enough, and $(x)^{1/2}$ otherwise.
For example, I write
$$
\left(\frac{1+\frac{9}{x^2}}{\sin \frac{\pi}{9}+2}\right)^{1/2}
\qquad\text{and not the ugly}\qquad
\sqrt{\frac{1+\frac{9}{x^2}}{\sin \frac{\pi}{9}+2}}
$$
Just as I write
$$
\exp\left(\frac{1+\frac{9}{x^2}}{\sin \frac{\pi}{9}+2}\right)
\qquad\text{and not the illegible}\qquad
e^{\frac{1+\frac{9}{x^2}}{\sin \frac{\pi}{9}+2}}
$$
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