HP50g simplifing a root

[Albert Chan]

I discovered a necessary condition to simplify cubic: ³√(A ± √R) = a ± √r
If we found a, we can get r = (A/a-a*a)/3

Simplify equations from find_cbrt(): 4a³ - A = (3a) ³√(A²-R)

This implied ³√(A²-R) must be integer !

Example, ³√(9416 - 4256√5)

A² - R = 9416² - 4256² * 5 = -1906624
c = ³√(A² - R) = -124        // cube root may be simplified

4a³ = A + 3*c*a
a = ³√(A/4 + (3c/4)*a)

With my Casio FX 115MS:

5 =
³√(2354 - 93 Ans
=       → 12.36167494
=       → 10.63945257
=       → 11.09160665
=       → 10.97648027 // locking to 11, try it
11 = ↑
=       → 11      // = a
9416 / Ans - Ans Ans
=       → 735    // = 3r
√(Ans / 3 / 5
=       → 7       // = b

Add the sign, ³√(9416 - 4256√5) = 11 - 7√5