Simplifying 3rd root of (24 * sqrt(3)) I have problems following a solution towards simplifying a given polynomial.
$$Polynomial: p(x)=x^5+{\sqrt 3}x^4+24{\sqrt 3}x^2+72x$$
the zeros of this function (Polynomial roots? English isn't my native language, so I don't know how to express the point(s) at which the function meets the X-axis) are:
$$x_0 = -{\sqrt 3} \\
x_1 = 0 \\$$
and the complex ones, which are calculated with what "remains" after polynomial division, drawing the 3rd root, etc.:
$$x=\sqrt[\Large 3]{-24\sqrt 3}$$
The first problem comes now. The next step, without explanation, simplifies the above to the following:
$$x=\sqrt[\Large 3]{8*\sqrt 3^3e^{\Large_{i\pi}}}$$
How is this done or rather what's the logic behind it? Especially the 8 that somehow was transformed from the 24.

The $8$ wasn't "transformed from" the $24$. It was factored: $24=8\cdot3=8\cdot\sqrt 3^2$, so $24\sqrt3=8\cdot\sqrt3^3$. The $-1$ became rewritten as $e^{\pi i}$.
