hp 50 g simplify solutions

[peacecalc]

Hello friends,

on hpcalc.org there exist a nice little libary which makes it possible, that the hp 50g is able to solve exact equations of 4th order.

For instance: \[ x^4 - 16 = 0 \]

Okay it is very easy and even the hp 50g is able to factorize it, but that is not the point. The program "polyroot" gives a list with four entries. One of the entries looks like:

\[ \frac{1}{6\cdot \sqrt[3]{\frac{8\cdot \sqrt{3}}{9}}}\cdot \sqrt{- \left(3+3\cdot i\cdot \sqrt{3}\right)\cdot {\sqrt[3]{\frac{8\cdot \sqrt{3}}{9}}}^{3}-\left(4-4\cdot i\cdot \sqrt{3}\right)\cdot \sqrt[3]{\frac{8\cdot \sqrt{3}}{9}}}\cdot \sqrt{6}-\frac{1}{6\cdot \sqrt[3]{\frac{8\cdot \sqrt{3}}{9}}}\cdot \sqrt{- \left(3-3\cdot i\cdot \sqrt{3}\right)\cdot {\sqrt[3]{\frac{8\cdot \sqrt{3}}{9}}}^{3}-\left(4+4\cdot i\cdot \sqrt{3}\right)\cdot \sqrt[3]{\frac{8\cdot \sqrt{3}}{9}}}\cdot \sqrt{6} \\ -\frac{1}{3\cdot \sqrt[3]{\frac{8\cdot \sqrt{3}}{9}}}\cdot \sqrt{3\cdot {\sqrt[3]{\frac{8\cdot \sqrt{3}}{9}}}^{3}-4\cdot \sqrt[3]{\frac{8\cdot \sqrt{3}}{9}}}\cdot \sqrt{3} \]

Holy cow! If you take the command "->NUM" you get the short and expected short answer, but not as a simplified symbolic answer. I test a lot of combinations of the commands "SIMPLIFY" "COLLECT" and with the special commands "EXPLN" and so on.
I hope there is somebody in this universe who knows the trick(s) for reducing such a
monster expression to a simple complex number.

Thank you in advance
peacecalc

[peacecalc]

Hello,

for completing my last post (where is the post from "gilles", I saw some hours ago???), HERE you can see the package I mean.

greetings
peacecalc

[Gerald H]

Simplifying in general is very difficult.

Wolfram Alpha can simplify

3*√66637+√(1554+6*√66637)-(√(777+6*√111)+√(600510-2*(3*√111)+2*√(465992541-1199466*√999)))

correctly although I haven't found an electronic calculator that can.

[Tugdual]

When I come across simplification questions, my first reaction is: How do you define "simple". Looks like there is not even consensus about that. May be a way to approach that would be to define the "energy" of an expression and seek for minimum but you're not certain the minimum would be "pleasant to read". The 50g actually defines a number of transformations instead of just a simplification.

[peacecalc]

Hello all,

thank you for your answers. Believe me, I don't want to start a philosophical disputation about: what is simple?

The equation x^4-16=0 has four simple solutions. So I thought as a simple mind, it might be possible that a CAS is able to reduce it to a simple complex number.

In my eyes an expression A is more simple than expression B, if A has less numbers and operators representing a solution then B (and it is supposed that then A is easier readable like B).

I'm not familiar with CAS programming, why such kind of simplifying (there are no variables at all) is for a CAS a hard piece of cake ?

greetings
peacecalc

[Marcus von Cube]

You could try RE and IM first to get the real and imaginary components of the solution.

[Gilles]

(08-10-2015 09:44 AM)peacecalc Wrote:  Hello all,

thank you for your answers. Believe me, I don't want to start a philosophical disputation about: what is simple?

The equation x^4-16=0 has four simple solutions. So I thought as a simple mind, it might be possible that a CAS is able to reduce it to a simple complex number.

In my eyes an expression A is more simple than expression B, if A has less numbers and operators representing a solution then B (and it is supposed that then A is easier readable like B).

I'm not familiar with CAS programming, why such kind of simplifying (there are no variables at all) is for a CAS a hard piece of cake ?

greetings
peacecalc

Hi

'X^4-16' SOLVEVX SIMPLIFY
returns the simplified result

With your result perhaps you can manipulate and simplify the equation "by part" in the equation writer, but it seems not obvious.

[Brad Barton]

'X^4-16' FACTOR SOLVEVX provides the exact solutions.

[peacecalc]

Hello all,

@Marcus: well it is a possibility and I get indeed a shorten expression (I tried it of course). But it has its limits it stops with nested roots ... and no further simplyfication was possible.

@Gilles: this simple case is of course solveable with the means of the hp 50g. It is a pity, that the general program (polyroot) isn't able to give the simple answers for the simple cases of equations. I tried your suggestion: simplification "by parts", too. But I failed with this approach. I even don't see what the pattern which is the obstacle for simplyfication (of course it seems to be the nested roots and nroots, but nothing more specific).

@Brad: please have a look to the answer @Gilles.