HP50g simplifing a root
09-30-2020, 02:22 AM (This post was last modified: 09-30-2020 02:42 AM by Albert Chan.)
Post: #3
 Albert Chan Senior Member Posts: 2,699 Joined: Jul 2018
RE: HP50g simplifing a root
For general case, to solve for all a, b:

﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ³√(n + m√k) = a + b√k

(a + b√k)³ = a³ + 3a²b√k + 3ab²k + b³k√k

n = a³ + 3ab²k ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ → n/a - a² = 3b²k
m = 3a²b + b³k ﻿ ﻿ ﻿ ﻿ ﻿ → 3m/b - 9a² = 3b²k

Equate the 2 to eliminate k, we have b = 3ma / (n + 8a³)

Substitute b to n = ... equation, we have a cubic equation, as function of A = a³

XCas> f(n, m, k) := horner([64, -48*n, 27*k*m^2-15*n^2, -n^3], A)

For first example, ³√(26 - 15√3):

XCas> expand(f(26,-15,3)) ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ → 64*A^3 - 1248*A^2 + 8085*A - 17576
XCas> proot(ans()) ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ →﻿ [5.75-1.125*i , 5.75+1.125*i , 8.0]

Above solved for A = a³. Each A produce 3 roots of a, thus a totaled 3×3 = 9 roots.
For this example, the real root is integer, a=2.

XCas> b := 3*m*a/(n + 8*a^3)
XCas> subst([a, b], [n,m,a] = [26,-15,2]) ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ → [2, -1]
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 Messages In This Thread HP50g simplifing a root - peacecalc - 09-29-2020, 09:22 PM RE: HP50g simplifing a root - Albert Chan - 09-29-2020, 11:47 PM RE: HP50g simplifing a root - Albert Chan - 09-30-2020 02:22 AM RE: HP50g simplifing a root - Albert Chan - 09-30-2020, 10:50 PM RE: HP50g simplifing a root - Albert Chan - 10-01-2020, 07:31 AM RE: HP50g simplifing a root - peacecalc - 09-30-2020, 05:33 AM RE: HP50g simplifing a root - peacecalc - 10-01-2020, 02:20 PM RE: HP50g simplifing a root - Albert Chan - 10-01-2020, 05:22 PM RE: HP50g simplifing a root - peacecalc - 10-04-2020, 06:05 PM RE: HP50g simplifing a root - Albert Chan - 10-04-2020, 11:48 PM RE: HP50g simplifing a root - peacecalc - 10-04-2020, 07:36 PM RE: HP50g simplifing a root - peacecalc - 10-05-2020, 11:36 AM RE: HP50g simplifing a root - Albert Chan - 10-05-2020, 05:01 PM RE: HP50g simplifing a root - peacecalc - 10-06-2020, 05:25 AM RE: HP50g simplifing a root - Albert Chan - 10-06-2020, 09:40 AM RE: HP50g simplifing a root - Albert Chan - 10-06-2020, 12:06 PM RE: HP50g simplifing a root - Albert Chan - 10-06-2020, 04:13 PM RE: HP50g simplifing a root - Albert Chan - 10-07-2020, 06:12 PM RE: HP50g simplifing a root - Albert Chan - 10-09-2020, 12:20 AM RE: HP50g simplifing a root - Albert Chan - 10-09-2020, 02:31 PM RE: HP50g simplifing a root - Albert Chan - 10-11-2020, 06:28 PM RE: HP50g simplifing a root - Albert Chan - 10-12-2020, 03:17 AM RE: HP50g simplifing a root - Albert Chan - 10-24-2020, 02:19 PM RE: HP50g simplifing a root - Albert Chan - 10-12-2020, 10:54 PM RE: HP50g simplifing a root - CMarangon - 10-12-2020, 11:45 PM RE: HP50g simplifing a root - grsbanks - 10-13-2020, 06:46 AM RE: HP50g simplifing a root - Albert Chan - 10-09-2020, 05:21 PM RE: HP50g simplifing a root - Albert Chan - 10-10-2020, 03:58 PM RE: HP50g simplifing a root - Albert Chan - 10-10-2020, 04:49 PM RE: HP50g simplifing a root - peacecalc - 10-12-2020, 08:49 PM RE: HP50g simplifing a root - peacecalc - 10-13-2020, 06:30 AM RE: HP50g simplifing a root - peacecalc - 10-13-2020, 06:36 AM

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