HP50g simplifing a root
09-30-2020, 10:50 PM (This post was last modified: 10-01-2020 05:09 AM by Albert Chan.)
Post: #5
 Albert Chan Senior Member Posts: 2,682 Joined: Jul 2018
RE: HP50g simplifing a root
(09-30-2020 02:22 AM)Albert Chan Wrote:  (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)

Instead of solving cubic, we can simply filter all (a,b), keeping only integers.
Since a divides n, just check divisors of n:

XCas> find_ab(n,m) := remove(x -> frac(x[1]), map(divisors(n).*sign(n) , a->[a, 3*m*a/(n+8*a^3)]))
XCas> find_abk(n,m) := map(find_ab(n,m), x -> x[0] + x[1] * sqrt((n/x[0]-x[0]^2)/(3*x[1]^2)))

XCas> find_abk(26,-15) ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ → $$[2 - \sqrt{3}]$$
XCas> find_abk(9416, -4256) ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ → $$[11 - 7 \sqrt{5}]$$

XCas> simplify((99+100*sqrt(101))^3) ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ → $$300940299 + 103940300\sqrt{101}$$
XCas> find_abk(300940299, 103940300) ﻿ ﻿ ﻿ ﻿ ﻿ → $$[99 + 100 \sqrt{101}]$$
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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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