HP50g simplifing a root

[Albert Chan]

Timings suggest it may be faster to solve the cubic, than factoring for divisors.
Again, we assumed n, m, k all integers.

Code:

find_cbrt(n,m,k):= { 
  local a;      // solved for A=a^3, then take cube root for a
  a := fsolve(horner([64,-48*n,-15*n*n+27*m*m*k,-n^3],x), x=n);
  a := exact(surd(a,3));
  if (remain(n,a)) return surd(n+m*sqrt(k),3);
  return a + sign(m) * sqrt((n/a-a*a)/3);
}

XCas> find_cbrt(300940299, 103940300, 101)       → \(99 + 100 \sqrt{101}\)

If simplified form search failed, it returned the cube root as-is.

XCas> find_cbrt(300940299, 103940300, 102)       → \(\left(300940299 + 103940300\cdot \sqrt{102}\right)^{\frac{1}{3}}\)