Understanding simple equality

Question

Looks like I got stucked. Could you please help me to understand that simple equality?

If $A\in\mathbb{N}$ and $B\in\mathbb{N}$ than why do we have

$$ (7^B \pmod{11})^A \pmod{11} = (7^A \pmod{11})^B \pmod{11} ? $$

Answer 1

Intuitively, ignore the $\pmod {11}$s and you have $(7^A)^B=7^{AB}=(7^B)^A$ The outer $\pmod {11}$ clearly is no trouble-you just have to convince yourself the inner one isn't either.