HP50g simplifing a root
10-24-2020, 02:19 PM
Post: #32
 Albert Chan Senior Member Posts: 2,679 Joined: Jul 2018
RE: HP50g simplifing a root
(10-11-2020 06:28 PM)Albert Chan Wrote:  We have ³√(36 + 20i√7) = (-3 + i√7) / (-1)^(2/3)

Mathematica gives many "simplifed" forms, here are 2 of them.

(-3+i√7) * (-1-i√3)/2 = (√21+3)/2 + i/2*(3√3-√7) = (√21+3)/2 + i/2*√(34-6√21)

I was curious, given √(34-6√21), how to "simplify" to (3√3-√7) ?
(10-09-2020 05:21 PM)Albert Chan Wrote:  $$\sqrt[3]{A ± \sqrt{R}} = a ± \sqrt{r} \quad﻿ ⇒ \quad a = \large\frac{\sqrt[3]{A+\sqrt{R}} \;+\; \sqrt[3]{A-\sqrt{R}}}{2}$$

We can use the same trick (if LHS is real, for both ±)

$$\sqrt{A ± B\sqrt{c d}} = a\sqrt{c} \,±\, b\sqrt{d} \quad﻿ ⇒ \quad a\sqrt{c} = \large\frac{\sqrt{A+B\sqrt{c d}}\;+\;\sqrt{A-B\sqrt{c d}}}{2}$$

lua> A, B, k = 34, -6, 21
lua> C = B * sqrt(k)
lua> mean = (sqrt(A+C) + sqrt(A-C))/2
lua> mean^2
27

27 = 3*3*3 = a*a*c. We try a=c=3, d=7, so just solve for b:

(3√3 + b√7)² = (27+7b²) + 6b√21 = 34 - 6√21

We have b = -1, thus √(34 - 6√21) = 3√3 - √7

It is nice that XCas do this automatically

XCas> simplify(√(34-6*√(21))) ﻿ ﻿ ﻿ ﻿ ﻿ → 3√3 - √7
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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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