How to show that $\sqrt[3]{-1+\sqrt{-7}}+\sqrt[3]{-1-\sqrt{-7}}$ is a real number at a time before the invention of complex numbers I have read this PDF from ocw.mit.edu about complex numbers. There is one interesting question: Imagine yourself at the time, when complex numbers had to be invented yet. How to show that $$\sqrt[3]{-1+\sqrt{-7}}+\sqrt[3]{-1-\sqrt{-7}}$$ is a real number.

I think it's a stretch to claim that this result can be proven without complex numbers, especially because of the presence of $\sqrt{-7}$. Anyway, here's a way to look at it. Let $A = \sqrt[3]{-1 + \sqrt{-7}}$ and $B = \sqrt[3]{-1 - \sqrt{-7}}$. If $t = A + B$, then $$t^3 = A^3 + B^3 + 3AB(A + B) = (-1 + \sqrt{-7}) + (-1 - \sqrt{-7}) + 3t\sqrt[3]{1 - (-7)} = -2 + 6t$$ So $t$ is a root of the cubic $x^3 - 6x + 2$. Let $f(x) = x^3 - 6x + 2$. Then $f(-3) = -7 < 0$, $f(0) = 2 > 0$, $f(1) = -3 < 0$, and $f(3) = 11 > 0$. Hence, by the intermediate value theorem, $f$ has three real roots. Therefore, $t$ is real.
