Your original expression is equivalent to determining how the following behaves for various conditions on n:
In[1]:= FullSimplify[(1 - Sqrt[2])^(2 n) (1 + Sqrt[2])^(2 n)]
Out[1]= (-1)^(2 n)
which, for integer n, is one. But you can also evaluate it for various other values of n. The 2n+1 case can be treated similarly:
In[2]:= FullSimplify[(1 - Sqrt[2])^(2 n + 1) (1 + Sqrt[2])^(2 n + 1)]
Out[2]= (-1)^(1 + 2 n)
In this case, if for example n is an integer, then the value is -1.
