In one proof I'm trying to figure out, I've arrived at a part where I am thinking about whether $x≡1$ and $y≡1$ imply $x^ay^b≡1$ for any real integers a and b. I know that in general, if $u≡v$ and $c≡d$, then $ub≡vd$. But what if one of $a$ and $b$ is negative? Let's say that $a≡1$. Can you prove that $a^k≡1$ for any integer k, even if k could be negative?
Thank you in advance.