Two formulas for calculating root of a complex number in a exponential form

[R A V E N]

$$z_k=\sqrt[n]{u}=\sqrt[n]{r}e^{i\left(\frac{\phi+2k\pi}{n}\right)}, k=0,1,2,...,n-1$$

and

$$z_k=\sqrt[n]{u}=\sqrt[n]{re}^\frac{\phi+2k\pi}{n}, k=0,1,2,...,n-1$$

Which one is incorrect (note that in the first, $$e$$ is out of the root)?

 

[Mentallic]

The second one is incorrect, and I would read that one as being $$\left(\sqrt[n]{re}\right)^\frac{\phi+2k\pi}{n}$$

When you have $$z_k^n=re^{i(\phi+2k\pi)}$$

then taking the n_th would yield $$z_k=\left(re^{i(\phi+2k\pi)}\right)^{\frac{1}{n}}=r^{\frac{1}{n}}e^{i(\phi+2k\pi)\frac{1}{n}}=\sqrt[n]{r}e^{i(\frac{\phi+2k\pi}{n})}$$