Wonderfully

2 = \sqrt [3]{2 + \frac {10}{3 \sqrt{3}} } + \sqrt [3]{2 - \frac {10}{3 \sqrt{3}} }

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,

1 = \sqrt [3]{2 + \sqrt{5} } + \sqrt [3]{2 - \sqrt{5}}

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

Amazingly (see Ref 2 below),

\sqrt [3]{2 \pm \sqrt{5} } = \frac {1 \pm \sqrt{5}}{2}

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

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

Cubic Formula