In chapter 2 of Mathematics: A Very Short Introduction, the author (Timothy Gowers) writes:
Two elementary rules about raising numbers to powers are the following.
E1 $a^1 = a$ for any real number $a$.
E2 $a^{m+n} = a^m \times a^n$ for any real number $a$ and any pair of natural numbers $m$ and $n$.
Then comes this example:
Let us write $x$ for the number $2^{3/2}$. Then $x \times x = 2^{3/2} \times 2^{3/2}$ which, by E2, is $2^{3/2+3/2} = 2^3 = 8$.
But $3/2$ isn't a natural number! A similar example follows involving a negative powers, but those aren't natural either.
If this is a mistake, it may seem minor - but I want to be sure I haven't missed something.
In fact, $e^{m+n} = e^m \times e^n$ for any pair of real numbers $m$ and $n$. And, $a^{m+n} = a^m \times a^n$ for any positive real number $a$ and any pair of real numbers $m$ and $n$.