74. Multiplication has the distributive property, or distributivity, with respect to addition
$[a(b+c)=a b+a c]$. Does exponentiation have the distributive property with respect to multiplication? Does multiplication have the distributive property with respect to exponentiation?
74. Let there be any two operations on numbers, which we will symbolically denote by the signs $\bigcirc$ and $\cup$ (to give our reasoning a general character, we deliberately chose signs not used in arithmetic, where each of the signs $\bigcirc$ and $\cup$ can represent any operation: addition, multiplication, subtraction, division, exponentiation, logarithmization). It is said that the operation $\bigcirc$ has the distributive property with respect to the operation $\cup$ if the equality $(a \smile b) \bigcirc c=(a \bigcirc c) \cup(b \bigcirc c)$ always holds. For example, we have: $(a+b) \times c=(a \times c)+(b \times c)$; this equality expresses the distributive property of multiplication with respect to addition. On the other hand, the equalities:

$$
\begin{array}{r}
(a \times b)+c=(a+c) \times(b+c) \\
\left(a^{b}\right) c=(a c)^{b c} \ldots
\end{array}
$$

are not valid; therefore, neither addition with respect to multiplication nor multiplication with respect to exponentiation have the distributive property. However, exponentiation does have this property with respect to multiplication, because $(a b)^{c}=a^{c} b^{c}$.