TITLE: Power Rule for group
QUESTION [5 upvotes]: Is it true that, for $x$ integer
$$(ab)^x=b^xa^x?$$
I believe the fact $$(ab)^x=(a^xb^x)$$ is not true in general, right?

REPLY [3 votes]: If you fix a particular value of $x$, though, there might be nonabelian groups where
$$
(a b)^x = a^x b^x\tag{pow}
$$
for all $a, b \in G$. For instance if $x$ is an odd prime, there exists a non-abelian group of order $x^3$ and exponent $x$, so (pow) holds in $G$ for such an $x$.