6. For natural numbers $a, b, c$, define the operation * such that it satisfies
$$
\begin{array}{l}
(a * b) * c=a *(b c), \\
(a * b)(a * c)=a *(b+c) .
\end{array}
$$

Then the value of $3 * 4$ is ( ).
(A) 12
(B) 64
(C) 81
(D) None of the above
6.C.

In $(a * b) * c=a *(b c)$, let $b=c=1$, we get $(a * 1) * 1=a * 1$,
so, $a * 1=a$.
In $(a * b)(a * c)=a *(b+c)$,
let $b=c=1$, we get $(a * 1)^{2}=a * 2$, so, $a * 2=a^{2} ;$
let $b=2, c=1$, we get $(a * 2)(a * 1)=a * 3$, so, $a * 3=a^{2} \cdot a=a^{3} ;$
let $b=3, c=1$, we get $(a * 3)(a * 1)=a * 4$, so, $a * 4=a^{3} \cdot a=a^{4}$.
Therefore, $3 * 4=3^{4}=81$.