15.35 Given the binary operation “ * ” defined as $a * b=a^{b}$ (where $a$ and $b$ are any positive numbers), then for all positive numbers $a, b, c, n$, we have
(A) $a * b=b * a$.
(B) $a *(b * c)=(a * b) * c$.
(C) $\left(a * b^{n}\right)=(a * n) * b$.
(D) $(a * b)^{n}=a *(b n)$.
(E) None of the above.
(21st American High School Mathematics Examination, 1970)
[Solution] Given $a * b=a^{b}$, and $b * a=b^{a}$, generally speaking, $a^{b}$ is not equal to $b^{a}$, so (A) is not true.
Also, $(a * b) * c=a^{b} * c=\left(a^{b}\right)^{c}=a^{b c}$,
while $a *(b * c)=a * b^{c}=a^{b^{c}}$, for all positive numbers, they are not necessarily equal, so (B) is not true.

Given $a * b^{n}=a^{b^{n}}$, and $(a * n) * b=a^{n} * b=a^{n b}$, then (C) is not true. Given $(a * b)^{n}=\left(a^{b}\right)^{n}=a^{b n}$, and $a *(b n)=a^{b n}$, so (D) is true. Therefore, (E) is not true.
Hence, the correct choice is (D).