The sum of $3$ real numbers is known to be zero. If the sum of their cubes is $e^{\pi}$ , what is their product equal to?

Question

The answer for this question is $ \frac{e^{\pi}}{3} $, and I don't understand why.

I tried to let the three real numbers be $a, b$, and $c$. This meant that $$ \begin{align} a + b + c &= 0 \\ a^3 + b^3 + c^3 &= e^{\pi}\end{align} $$

How do we get the value of $abc$ from the two equations above? I tried cubing the first equation but there are a lot of other terms in the expansion that seem to be unnecessary. Any help would be greatly appreciated.

Answer 1

Using $c = -(a+b),$

$$e^\pi = a^3+b^3-(a+b)^3 = -3(a^2 b+b^2 a)=-3ab(a+b) = 3abc$$