Wonderfully

Why? Think of the cubic formula. Is the line above an instance of the latter and, if so, what equation does it solve?

The answer is quite easy: the formula shows a solution to x^3 - 2x - 4 = 0 . Directly by inspection, this equation has a single real root x = 2 . Since the expression on the right is verifiably real, the identity holds.

Similarly,

which comes from the application of the cubic formula to the equation x^3 + 3x - 4 = 0 .

Amazingly (see Ref 2 below),

So that, while \sqrt [3]{2} appears simpler than \sqrt [3]{2 \pm \sqrt{5} } , the latter is constructible while the former is not.

### References

- R. I. Hess,
*Puzzles from Around the World*, in*The Mathemagician and Pied Piper*, A K Peters, 1999, pp. 57 and 68 - T. J. Osler, Cardan Polynomials and the Reduction of Radicals,
*Math Magazine*, Vol. 74, No. 1 (Feb., 2001), pp. 26-32