How do we compute $\sqrt[3]{x_1} +\sqrt[3]{x_2} $ using the fact that $x_1 + x_2 = 4 , x_1x_2 = -1$? Given quadratic equation
$$x^2 -4x-1 = 0$$
How do we compute $\sqrt[3]{x_1} +\sqrt[3]{x_2} $ using the fact that $x_1 + x_2 = 4 , x_1x_2 = -1$? 
Regards

Using $a^3+b^3= (a+b)(a^2-(ab)+b^2)$:
$(x_1+x_2)=$
$(x_1^{1/3}+x_2^{1/3})(x_1^{2/3} -(x_1x_2)^{1/3}+x_2^{2/3}).$
Set $X = x_1^{1/3}+x_2^{1/3}$
$4=X(x_1^{2/3}+x_2^{2/3}+1).$
Express $x_1^{2/3}+x_2^{2/3}$ in terms of $X$:
$X^2= (x_1^{1/3}+x_2^{1/3})^2=$ 
$x_1^{2/3}+x_2^{2/3} +2x_1^{1/3}x_2^{1/3}$;
$X^2+2= x_1^{2/3}+x_2^{2/3}.$
Finally:
$4=X(X^2+3)$.
$X^3+3X -4=0.$
$X=1$ is a solution, factoring:
$(X-1)(X^2 +X+4)=0$.
Any other solutions?
