smc

Let $a=2,b=3,c=n=4$. If all of them are false, the answer must be $E$. If one does not fail, we will try to prove it. For option $A$, we have $2^3=3^2$, which is clearly false. For option $B$, we have $2^{81}=8^4$, which is false. For option $C$, we have $2^{81}=16^3$, which is false. For option $D$, we have $8^4=2^{12}$, which is true. The LHS is $(a^b{)}^n$. By the elementary definition of exponentiation, this is $a^b$ multiplied by itself $n$ times. Since each $a^b$ is actually $a$ multiplied $b$ times, the expression $(a^b{)}^n$ is $a$ multiplied by itself $bn$ times. The RHS is $a^{bn}$. This is $a$ multiplied by itself $bn$ times. Thus, the LHS is always equal to the RHS, so $\boxed{(a\star b{)}^n=a\star (bn)}$ is the only correct statement.