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Why does 8^7 ≡ (-5)^7 (mod 13) and (25)^3 ≡ (-1)^3 (mod 13)?

In essence I want to show that 8^7 + 5^7 = 13^7, so that both sides of the equation ≡ 0 (mod 13) and therefore 8^7 ≡ (-5)^7 (mod 13).

I know that in a field of characteristic p>0, (x + y)^p = x^p + y^p, but the problem here is that the exponent is 7, not 13.

I also noticed that the equation works for odd integers but not even ones, i.e., 8^3 ≡ (-5)^3 (mod 13), 8^5 ≡ (-5)^5 (mod 13), etc.

Can anyone help me out here? Thanks.