Supposing $S=x_1+x_2$ and $P=x_1x_2$ where $x_1$ and $x_2$ are the roots of the quadratic equation $ax^2+bx+c=0$ (it's clear the equation is equivalent to $x^2-Sx+P=0$ where $S=-b/a$ and $P=c/a$), how we can calculate $(x_1^{x_2})(x_2^{x_1})$?
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Use the quadratic equation to get the roots in terms of $S$ and $P$. But there's no point of doing this if you already know the roots $x_1,x_2$. I'd be interested if there were anything simple about the value of $x_1^{x_2}x_2^{x_1}$ expressed in terms of $S,P$. – coffeemath Nov 28 '13 at 08:36