The 1/5 power in Mathematica is defined as x^(1/5) == Exp[(1/5) Log[x]]. When x is negative, the principal value of the logarithm has I Pi as imaginary part, so that
In[9]:= (-1)^(1/5) == Exp[(1/5) Log[-1]]
Out[9]= True
and
In[10]:= Exp[(1/5) Log[-1]] // N
Out[10]= 0.809017 + 0.587785 I
This issue is an endless source of confusion and frustration for beginners, which had been raised in a real-centered theory of roots.
