elementary number theory - Reducing products in modular arithmetic

During an effort to show that $2^{20} \equiv 1 \mod{41}$, I have done the following:

$2^{20} = \left(2^5\right)^4 = 32^4$

Since $32 \equiv -9 \mod{41}$, we get $32^4 \equiv (-9)^4 = 81\cdot81 \mod 41$

From here, I know that I can reduce the 81s, such that I get $2^{20} \equiv (-1)(-1) \mod 41$, so I can solve the problem, but I can't connect this reduction to a particular rule of modular arithmetic.

Question

From $2^{20} \equiv 81 \cdot 81 \mod 41$, which rule is it that states that the $81$s can be reduced to their individual congruences, modulo $41$? In other words, why may I reduce them to $(-1)(-1)$?

I'm familiar with some of the rules, like the basic addition/subtraction/multiplication/power ones, but if it's one of these, I don't quite see the connection.