HP50g simplifing a root
10-06-2020, 09:40 AM
Post: #15
 Albert Chan Senior Member Posts: 2,699 Joined: Jul 2018
RE: HP50g simplifing a root
(10-06-2020 05:25 AM)peacecalc Wrote:  For calculating A and B I use k as a given value, hmm is that wrong? Are there possibly other solutions with another k as given? But when this would be possibly then we have a lot of more solutinon, too much...?

It is possible for the same A,B, but different k's, we can simplify the cube roots.
(Actually, I am not so sure. Seems pretty rare for this to happen ...)

I know that for same A,k, but different B's, it is possible (at least, if we included imaginary numbers)

Below searched for all A, such that ³√(A ± √R) = a ± √r, ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ where R = B²k, r = b²k

r = b²k = (A/a - a*a)/3 = (x-y)/3, where x=A/a, y=a*a
B = b * (3a² + b²k)
R = B²k = r * (3a² + r)^2 = (x-y)/3 * (3y + (x-y)/3)² = (x-y)/27*(x+8*y)²

XCas> getR(x,y) := (x+8*y)^2*(x-y)/27
XCas> find_all_a(A) := remove(a-> denom(getR(A/a,a*a))!=1, divisors(A).*sign(A))
XCas> find_all_abk(A) := map(a -> a + sqrt((A/a-a*a)/3), find_all_a(A))

For A=90, k=7, we have 2 cube roots able to simplify, B = 34, 101i
Note: because of "±", we also have the "-" solutions.

XCas> s := find_all_abk(90)﻿ ﻿ → $$[\;3+\sqrt7,\quad\quad 6+i\sqrt7,\quad\;\; 15+i\sqrt{73}, \quad\quad 30+i\sqrt{299}]$$
XCas> simplify(s .^ 3) ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ → $$[90+34\sqrt7,\; 90+101i\sqrt7,\; 90+602i\sqrt{73}, \;90+2401i \sqrt{299}]$$
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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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