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Re: simplification

• To: mathgroup at smc.vnet.net
• Subject: [mg121847] Re: simplification
• From: "Dr. Wolfgang Hintze" <weh at snafu.de>
• Date: Wed, 5 Oct 2011 03:58:56 -0400 (EDT)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
• References: <j6e693$kef$1@smc.vnet.net>

"dimitris" <dimmechan at yahoo.com> schrieb im Newsbeitrag 
news:j6e693$kef$1 at smc.vnet.net...
> Hello.
>
> Let
>
> o1 = 1 + Sqrt[15 + 2*Sqrt[35] + 2*Sqrt[6*(6 + Sqrt[35])]];
> o2 = 1 + Sqrt[3] + Sqrt[5] + Sqrt[7];
>
> o1 is equal to o2.
>
> o1 == o2 // FullSimplify
> True
>
> The question is how to make Mathematica to simplify o1 to o2.
>
> Thanks
> Dimitris
>
There are even simpler examples which seem to bring Mathematica to its 
limits very soon.

Consider this sequence
In[95]:=
Simplify[Sqrt[1 + a + 2*Sqrt[a]], {a > 0}]
Out[95]=
1 + Sqrt[a]


In[96]:=
Simplify[Sqrt[a^2 + b + 2*a*Sqrt[b]], {a > 0, b > 0}]
Out[96]=
a + Sqrt[b]


In[97]:=
Simplify[Sqrt[a + b + 2*Sqrt[a*b]], {a > 0, b > 0}]
Out[97]=
Sqrt[a + b + 2*Sqrt[a*b]]

Maybe Mathematica does not like radicals. If we help a little bit, it 
works out fine

In[126]:=
Simplify[Sqrt[a + b + 2*Sqrt[a*b]] /. {a -> u^2, b -> v^2}, {u > 0, v > 
0}] /. {u -> Sqrt[a], v -> Sqrt[b]}
Out[126]=
Sqrt[a] + Sqrt[b]

The example o1 provided can be treated by Factor using the option 
Extension like this ...

In[140]:=
f = Factor[Sqrt[15 + 2*Sqrt[35] + 2*Sqrt[6*(6 + Sqrt[35])]], 
Extension -> {Sqrt[3], Sqrt[5], Sqrt[7]}]
Out[140]=
Root[3481 - 3180*#1^2 + 782*#1^4 - 60*#1^6 + #1^8 & , 8]
In[141]:=
N[%]
Out[141]=
6.613870096133257

... but this seems to lead to nothing useful.

Wolfgang