prouve :
Hi
If you consider the equation $\displaystyle x^3+5x-6=0$ you can find that x=1 is the only real solution since $\displaystyle x^3+5x-6=(x-1)(x^2+x+6)$
Now if you try to solve it using Cardan method Méthode de Cardan - Wikipédia you will find that one real solution is
$\displaystyle \sqrt[3]{3+\sqrt{9+\frac{125}{27}}}-\sqrt[3]{-3+\sqrt{9+\frac{125}{27}}}$
Since 1 is the only real solution you get $\displaystyle \sqrt[3]{3+\sqrt{9+\frac{125}{27}}}-\sqrt[3]{-3+\sqrt{9+\frac{125}{27}}} = 1$