Bad, with Mathematica 10.0.0 the sequence of evaluation does not matter seemingly, needs investigation. Sometimes the second expression gets through, mostly not. RootApproximant[] works constantly
In[1]:= FullSimplify[((-3965 + 31 Sqrt[1054]) (-1 +
9 Sqrt[1254/(230348 - 1891 Sqrt[1054] +
15710428 Sqrt[17/(341575 - 8174 Sqrt[1054])] -
10588825 Sqrt[62/(341575 - 8174 Sqrt[1054])])]))/22572 == 0]
Out[1]= True
done as the second one, works always ...
In[1]:= FullSimplify[((51250980 Sqrt[2] + 777480 Sqrt[527] +
Sqrt[31 (341575 - 8174 Sqrt[1054])] (4087 +
31 Sqrt[1054]))/(4087 + 31 Sqrt[1054]) -
9 Sqrt[2] (1848 +
Sqrt[(627 (-341575 + 8174 Sqrt[1054]))/(-94972831936 +
2528782877 Sqrt[1054] -
10732578888200 Sqrt[17/(341575 - 8174 Sqrt[1054])] +
5799967553399 Sqrt[
62/(341575 - 8174 Sqrt[1054])])] (-4092 Sqrt[2] +
Sqrt[31 (341575 - 8174 Sqrt[1054])])))/22572 == 0]
done first, no result, kernel hangs up, the simpler one on top gets again In[1] ...
In[2]:= RootApproximant[((51250980 Sqrt[2] + 777480 Sqrt[527] +
Sqrt[31 (341575 - 8174 Sqrt[1054])] (4087 +
31 Sqrt[1054]))/(4087 + 31 Sqrt[1054]) -
9 Sqrt[2] (1848 +
Sqrt[(627 (-341575 + 8174 Sqrt[1054]))/(-94972831936 +
2528782877 Sqrt[1054] -
10732578888200 Sqrt[17/(341575 - 8174 Sqrt[1054])] +
5799967553399 Sqrt[
62/(341575 - 8174 Sqrt[1054])])] (-4092 Sqrt[2] +
Sqrt[31 (341575 - 8174 Sqrt[1054])])))/22572]
Out[2]= 0