I found a beautiful form of the sum of square roots.

$a,b,cinmathbb{R}$

begin{eqnarray*}

A &:=& abc\

B &:=& a+b+c\

C &:=& ab+bc+ca

end{eqnarray*}

$$sqrt{a}+sqrt{b}+sqrt{c} = sqrt{A^0B+2sqrt{A^0C+2sqrt{A^1B+2sqrt{A^2C+2sqrt{A^5B+2sqrt{A^{10}C}}}}}} cdotscdots$$

This property is based on the next relation.

$$(sqrt{a}+sqrt{b}+sqrt{c})^2 = a+b+c+2(sqrt{ab}+sqrt{bc}+sqrt{ca})$$

$$(sqrt{ab}+sqrt{bc}+sqrt{ca})^2 = ab+bc+ca+2(sqrt{ab^2c}+sqrt{abc^2}+sqrt{a^2bc})$$

$$(sqrt{ab^2c}+sqrt{abc^2}+sqrt{a^2bc})^2 = ab^2c+abc^2+a^2bc+2(sqrt{a^2b^3c^3}+sqrt{a^3b^2c^3}+sqrt{a^3b^3c^2})$$

$$vdots$$

In this way, three sums of square roots generate three sums of square roots. Therefore, it’s a recursive structure.

$$$$

The same property can be confirmed for the cube roots.

$a,binmathbb{R}$

begin{eqnarray*}

A &:=& ab\

B &:=& a+b

end{eqnarray*}

$$sqrt[3]{a}+sqrt[3]{b} = sqrt[3]{A^0B+3sqrt[3]{A^1B+3sqrt[3]{A^4B+3sqrt[3]{A^{13}B+3sqrt[3]{A^{40}B+3sqrt[3]{A^{121}B}}}}}} cdotscdots$$

Please let me know if you have any interesting information related to these.

Mathematics Asked by isato on November 21, 2021

1 AnswersI don't know why, but it seems that the exponent of A in the case of the cube root is in this sequenceA003462.

Answered by ratbase on November 21, 2021

4 Asked on August 5, 2020 by dharmendra-singh

1 Asked on August 4, 2020 by w-mu

differential geometry geodesic riemannian geometry smooth manifolds

1 Asked on August 4, 2020 by hash-man

1 Asked on August 3, 2020 by ifn47

2 Asked on August 3, 2020 by 2132123

1 Asked on August 3, 2020 by user

calculus of variations euler lagrange equation functional analysis reference request

1 Asked on August 2, 2020

cyclotomic polynomials elementary number theory primality test prime numbers

1 Asked on July 31, 2020 by twentyyears

1 Asked on July 30, 2020 by pea-terzi

elementary number theory examples counterexamples prime numbers

2 Asked on July 28, 2020 by lucas

0 Asked on July 27, 2020 by strictly_increasing

numerical calculus numerical methods ordinary differential equations probability theory stochastic differential equations

1 Asked on July 27, 2020 by gulzar

2 Asked on July 26, 2020 by outlier

Get help from others!

Recent Answers

- Harper - Reinstate Monica on How can I repair a split copper pipe in place?
- d.george on How can I repair a split copper pipe in place?
- FreeMan on How can I repair a split copper pipe in place?
- Quoc Vu on Safe to put tiles on top of two 3/4″ fir board layers & Ditra-Heat?
- Vincent on Safe to put tiles on top of two 3/4″ fir board layers & Ditra-Heat?

© 2021 InsideDarkWeb.com. All rights reserved.