Mathematica the Simplify[] command

[Mentallic]

Homework Statement

Simplify to a certain extent, as much as possible (factorized form would be best):

$$\frac{\sqrt{3}}{2}\left(\frac{md}{m-\sqrt{3}}\right)^2+\frac{\sqrt{3}}{2}\left(\frac{m(s+d)}{m+\sqrt{3}}\right)^2+\frac{1}{2}\left(s+\frac{md}{m-\sqrt{3}}-\frac{m(s+d)}{m+\sqrt{3}}\right)\left(\frac{\sqrt{3}md}{\sqrt{3}-m}+\frac{\sqrt{3}m(s+d)}{m+\sqrt{3}}\right)$$

The Attempt at a Solution

This isn't an actual question but rather a part of a bigger picture. I need this simplified as much as possible before I can even think of manipulating it further.

I've tried quite a bit, but after expanding the last factored expression, it became pretty ugly. This route seems most promising:

$$\frac{\sqrt{3}}{2}\left(\left(\frac{md}{m-\sqrt{3}}\right)^2+\left(\frac{m(s+d)}{m+\sqrt{3}}\right)^2+\left(s+\frac{md}{m-\sqrt{3}}-\frac{m(s+d)}{m+\sqrt{3}}\right)\left(\frac{m(s+d)}{m+\sqrt{3}}-\frac{md}{m-\sqrt{3}}\right)\right)$$

$$\frac{\sqrt{3}}{2}\left(\left(\frac{md}{m-\sqrt{3}}\right)^2+\left(\frac{m(s+d)}{m+\sqrt{3}}\right)^2+s\left(\frac{m(s+d)}{m+\sqrt{3}}-\frac{md}{m-\sqrt{3}}\right)-\left(\frac{2\sqrt{3}md+\sqrt{3}ms-m^2s}{m^2-3}\right)^2\right)$$

$$\frac{\sqrt{3}}{2}\left(\left(\frac{md}{m-\sqrt{3}}\right)^2+\left(\frac{m(s+d)}{m+\sqrt{3}}\right)^2-s\left(\frac{2\sqrt{3}d+\sqrt{3}s-ms}{m^2-3}\right)-m^2\left(\frac{2\sqrt{3}d+\sqrt{3}s-ms}{m^2-3}\right)^2\right)$$

I'm unsure of what else to do without making a big mess. And this still isn't simple enough for what I need this for, so if you have the stomach to tackle this, please do so

 

[icystrike]

 

[Mentallic]

Thanks for the response.

I don't quite understand what your final solution is. When it gets to the : and then it has just one entire fraction. That one fraction can't possibly be equal? It only has a second degree for m, while if you look at this part which is located on the last line in the last factor of my OP:

$$m^2\left(\frac{2\sqrt{3}d+\sqrt{3}s-ms}{m^2-3}\right)^2$$

it clearly suggests the denominator should be of a 4th degree in m, no?

 

[icystrike]

I hope this answers your question.(=

 

[Mentallic]

Oh wow thanks a lot for that!

It still took me a while to understand what was happening, but it all sunk in and when I tested it for some values of s,d and m and they turned out equal, I was even more ecstatic!

 

[icystrike]

you are most welcome(=

 

[Mentallic]

Did you simplify this yourself, or did you use a program of some sort?
Reason being because I'm going to need more simplifying of this type to be done.

 

[icystrike]

I've checked with a program and indeed it is the simplest form.

 

[Feldoh]

If you have access to Mathematica the Simplify[] command would come in handy.

Come to think of it there might also be something similar you can use on Wolfram Alpha to simplify expressions.