Are we still looking for the fabled case where neither √A nor √B denests, but √A+√B does? Unfortunately, this isn't a clear cut case, because the denesters go off the rails instead of declaring impossibility. Even the numerics is shaky. Out[19]= Sqrt[5] + Sqrt[7] == Sqrt[4 + (-54 + 35 I Sqrt[3])^(1/3)/3^(2/3) + 13/(3 (-54 + 35 I Sqrt[3]))^(1/3)] + \[Sqrt]((1/(162 I + 105 Sqrt[3]))(1296 I + 840 Sqrt[3] - 35 3^(5/6) (-54 + 35 I Sqrt[3])^(1/3) - 54 I (3 (-54 + 35 I Sqrt[3]))^(1/3) + 13 I (3 (-54 + 35 I Sqrt[3]))^(2/3))) Try to denest this RHS with Swastik's new package: In[20]:= ResourceFunction["RadicalDenest"] /@ %[[2]] During evaluation of In[20]:= Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information. During evaluation of In[20]:= Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information. Out[20]= Sqrt[ 4 + (-54 + 35 I Sqrt[3])^(1/3)/3^(2/3) + 13/(3 (-54 + 35 I Sqrt[3]))^( 1/3)] + \[Sqrt]((1/( 162 I + 105 Sqrt[3]))(1296 I + 840 Sqrt[3] - 35 3^(5/6) (-54 + 35 I Sqrt[3])^(1/3) - 54 I (3 (-54 + 35 I Sqrt[3]))^(1/3) + 13 I (3 (-54 + 35 I Sqrt[3]))^(2/3))) And with Corey's original package: In[49]:= $MaxExtraPrecision = 999; Strad /@ %20 During evaluation of In[49]:= N::meprec: Internal precision limit $MaxExtraPrecision = 999.` reached while evaluating -(105/13) Cos[2/3 (\[Pi]-ArcTan[Times[<<2>>]])]+3 Sqrt[13] Sin[1/3 (\[Pi]-ArcTan[Times[<<2>>]])]-54/13 Sqrt[3] Sin[2/3 (\[Pi]-ArcTan[Times[<<2>>]])]. During evaluation of In[49]:= MinimalPolynomial::nalg: (1/(Sqrt[6] 13^(1/4)))Cos[1/6 (\[Pi]+2 ArcTan[Times[<<2>>]])]^(\[Pi]/2+ArcTan[35/(18 Sqrt[3])]) Sqrt[(26 Sqrt[13]+<<1>>+(67+35 I Power[<<2>>]) Sin[1/3 <<6>>[<<1>>]]) <<1>>] is not an explicit algebraic number. What's all this trigamarole? In[50]:= %20 == \[Sqrt]5 + \[Sqrt]7 Out[50]= Sqrt[ 4 + (-54 + 35 I Sqrt[3])^(1/3)/3^(2/3) + 13/(3 (-54 + 35 I Sqrt[3]))^( 1/3)] + \[Sqrt]((1/( 162 I + 105 Sqrt[3]))(1296 I + 840 Sqrt[3] - 35 3^(5/6) (-54 + 35 I Sqrt[3])^(1/3) - 54 I (3 (-54 + 35 I Sqrt[3]))^(1/3) + 13 I (3 (-54 + 35 I Sqrt[3]))^(2/3))) == Sqrt[5] + Sqrt[7] In[51]:= N@% Out[51]= False Well, no wonder it didn't denest! Wait a minute, In[52]:= N[%%, 22] Out[52]= True That was a false negative! —rwg