HP50g simplifing a root

[peacecalc]

Hello folks,

I was wondering why my good old hp50g is not able to simplify:

\[ \left(26 - 15\cdot\sqrt{3} \right)^{\frac{1}{3}} = 2 - \sqrt{3} \]

Hmm.. it is indeed not trivial: the machine had to calculate:

\[ \left(26 - 15\cdot\sqrt{3} \right) = (a - b)^{3} = a^3 -3a^2b + 3ab^2 - b^3 \]

We first see that b must contain the root of three as a factor. And 26 must contain the cube of a:

So we get: \[ 26 = a^3 + 3ab^2 \] Why this? the two terms on the right side may not contain a root. So the other two term with a root should be: \[ - 15\sqrt{3} = - 3a^2b - b^3 \]

After a little guess we get: \[ 26 = 8 + 18 = 2^3 + 18 \] and \[ 18 = 3ab^2 \] We set a = 2 and for b we get the root of 3 out of the last equation.

Now it was clear why the calculator isn't able to transform the terms, there is no simple recipe for this. Look at the more complicate expression for simplification:

\[ \left( 9416 - 4256\sqrt{5} \right)^{\frac{1}{3}} = ? \]